{"id":1201,"date":"2026-01-20T18:54:30","date_gmt":"2026-01-20T08:54:30","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1201"},"modified":"2026-01-31T11:46:41","modified_gmt":"2026-01-31T01:46:41","slug":"year11-math-4-1-2-vectors-in-the-plane","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1201","title":{"rendered":"Year11- MATH-4-1-2 Vectors in the Plane"},"content":{"rendered":"\n

Welcome to Year 11! If you\u2019re diving into Specialist Mathematics or Mathematical Methods in Queensland (QCAA curriculum), vectors are about to become your new best friends.<\/p>\n\n\n\n

Think of a vector as a mathematical instruction<\/strong>. Until now, you\u2019ve mostly dealt with “scalars”\u2014numbers that only tell you how much<\/em> (like 5 kg of apples or a temperature of 25\u00b0C). Vectors are cooler because they tell you two things at once: how far<\/strong> and in what direction<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

1. The Basics: Magnitude and Direction<\/h3>\n\n\n\n

In the Cartesian plane (your standard x<\/mi>x<\/annotation><\/semantics><\/math> and y<\/mi>y<\/annotation><\/semantics><\/math> grid), a vector is represented as an arrow.<\/p>\n\n\n\n
    \n
  • Magnitude:<\/strong> This is the length of the arrow (how far).<\/li>\n\n\n\n
  • Direction:<\/strong> This is where the arrow is pointing.<\/li>\n<\/ul>\n\n\n\n

    Imagine you are telling a friend how to get from the school gate to the tuckshop. You wouldn’t just say “walk 50 meters.” You’d say “walk 50 meters East<\/strong>.” That\u2019s a vector.<\/p>\n\n\n\n

    \n\n\n\n

    2. How We Write Them (The Notation)<\/h3>\n\n\n\n

    Queensland teachers are big on correct notation. You\u2019ll see vectors written in a few ways:<\/p>\n\n\n\n

      \n
    • Bold letters:<\/strong> \ud835\udc2e<\/mi>\\mathbf{u}<\/annotation><\/semantics><\/math> or \ud835\udc2f<\/mi>\\mathbf{v}<\/annotation><\/semantics><\/math> (common in textbooks).<\/li>\n\n\n\n
    • Arrow notation:<\/strong> A<\/mi>B<\/mi><\/mrow>\u2192<\/mo><\/mover>\\vec{AB}<\/annotation><\/semantics><\/math> (meaning the path from point A<\/mi>A<\/annotation><\/semantics><\/math> to point B<\/mi>B<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
    • Column vectors:<\/strong> [<\/mo>x<\/mi><\/mtd><\/mtr>y<\/mi><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>\\begin{bmatrix} x \\\\ y \\end{bmatrix}<\/annotation><\/semantics><\/math>. This is like a GPS coordinate. The top number tells you how many units to move horizontally, and the bottom tells you how many to move vertically.<\/li>\n\n\n\n
    • Unit Vector Form:<\/strong> x<\/mi>\ud835\udc22<\/mi>+<\/mo>y<\/mi>\ud835\udc23<\/mi><\/mrow>x\\mathbf{i} + y\\mathbf{j}<\/annotation><\/semantics><\/math>.\n
        \n
      • \ud835\udc22<\/mi>\\mathbf{i}<\/annotation><\/semantics><\/math> is one step in the x<\/mi>x<\/annotation><\/semantics><\/math> direction.<\/li>\n\n\n\n
      • \ud835\udc23<\/mi>\\mathbf{j}<\/annotation><\/semantics><\/math> is one step in the y<\/mi>y<\/annotation><\/semantics><\/math> direction.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
        \n

        Example:<\/strong> If \ud835\udc2f<\/mi>=<\/mo>3<\/mn>\ud835\udc22<\/mi>+<\/mo>2<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{v} = 3\\mathbf{i} + 2\\mathbf{j}<\/annotation><\/semantics><\/math>, you go 3 steps right and 2 steps up.<\/p>\n<\/blockquote>\n\n\n\n
        \n\n\n\n

        3. Vector “Math” (It’s easier than it looks)<\/h3>\n\n\n\n

        You can do arithmetic with vectors, but it\u2019s more visual than regular algebra.<\/p>\n\n\n\n

        Adding Vectors (The “Tip-to-Tail” Rule)<\/h4>\n\n\n\n

        To add vector \ud835\udc1a<\/mi>\\mathbf{a}<\/annotation><\/semantics><\/math> and vector \ud835\udc1b<\/mi>\\mathbf{b}<\/annotation><\/semantics><\/math>, you place the “tail” of \ud835\udc1b<\/mi>\\mathbf{b}<\/annotation><\/semantics><\/math> at the “tip” of \ud835\udc1a<\/mi>\\mathbf{a}<\/annotation><\/semantics><\/math>. The result (the resultant vector<\/strong>) is the shortcut from the very start to the very end.<\/p>\n\n\n\n

        Scalar Multiplication<\/h4>\n\n\n\n

        If you multiply a vector by a regular number (a scalar), you just change its length.<\/p>\n\n\n\n

          \n
        • 2<\/mn>\ud835\udc2f<\/mi><\/mrow>2\\mathbf{v}<\/annotation><\/semantics><\/math> is twice as long as \ud835\udc2f<\/mi>\\mathbf{v}<\/annotation><\/semantics><\/math>, pointing the same way.<\/li>\n\n\n\n
        • \u2212<\/mo>1<\/mn>\ud835\udc2f<\/mi><\/mrow>-1\\mathbf{v}<\/annotation><\/semantics><\/math> is the same length, but pointing in the exact opposite direction<\/strong>.<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          4. Calculating the Magnitude (Size)<\/h3>\n\n\n\n

          Since vectors on a plane form a right-angled triangle with the x<\/mi>x<\/annotation><\/semantics><\/math> and y<\/mi>y<\/annotation><\/semantics><\/math> axes, we use our old friend Pythagoras<\/strong> to find the length.<\/p>\n\n\n\n

          For a vector \ud835\udc2f<\/mi>=<\/mo>[<\/mo>x<\/mi><\/mtd><\/mtr>y<\/mi><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mrow>\\mathbf{v} = \\begin{bmatrix} x \\\\ y \\end{bmatrix}<\/annotation><\/semantics><\/math>, the magnitude (written as |<\/mi>\ud835\udc2f<\/mi>|<\/mi><\/mrow>|\\mathbf{v}|<\/annotation><\/semantics><\/math>) is:<\/p>\n\n\n\n

          |<\/mi>v<\/mtext>|<\/mi><\/mrow>=<\/mo>x<\/mi>2<\/mn><\/msup>+<\/mo>y<\/mi>2<\/mn><\/msup><\/mrow><\/msqrt><\/mrow>\\mathbf{|\\text{v}|} = \\sqrt{x^2 + y^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
          \n\n\n\n

          5. Why do we bother?<\/h3>\n\n\n\n

          In the Queensland syllabus, you\u2019ll use these to solve real-world problems involving:<\/p>\n\n\n\n

            \n
          1. Velocity:<\/strong> A plane flying North at 500 km\/h while a wind blows East at 50 km\/h.<\/li>\n\n\n\n
          2. Force:<\/strong> Two people pulling a rope in different directions.<\/li>\n\n\n\n
          3. Geometry:<\/strong> Proving things about shapes using midpoints and diagonals.<\/li>\n<\/ol>\n\n\n\n

            Pro-tip from a friendly AI:<\/strong> Always draw a diagram. Even if you think you see the answer in your head, drawing the arrows helps prevent those “silly” direction mistakes that haunt Year 11 exams.<\/p>\n\n\n\n

            \n\n\n\n

            6. Practice problem on how to find the angle (direction) of a vector using trigonometry.<\/h3>\n\n\n\n

            Great choice! In the Queensland curriculum, finding the “direction” of a vector usually means finding the angle \u03b8<\/mi>\\theta<\/annotation><\/semantics><\/math> it makes with the positive <\/strong>x<\/mi>x<\/annotation><\/semantics><\/math>-axis<\/strong> (going anticlockwise), or sometimes expressed as a bearing<\/strong>.<\/p>\n\n\n\n

            Let’s look at a classic exam-style problem.<\/p>\n\n\n\n

            \n\n\n\n

            The Problem<\/h4>\n\n\n\n

            Find the direction of the vector \ud835\udc2f<\/mi>=<\/mo>4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{v} = 4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

            Step 1: Sketch it<\/h4>\n\n\n\n

            Before touching your calculator, visualize the vector. Since both components are positive, the vector moves 4 units right<\/strong> and 3 units up<\/strong>. This puts us in the first quadrant<\/strong>.<\/p>\n\n\n\n

            Step 2: Identify your triangle<\/h4>\n\n\n\n

            By drawing a line from the tip of the vector down to the x<\/mi>x<\/annotation><\/semantics><\/math>-axis, you create a right-angled triangle where:<\/p>\n\n\n\n
              \n
            • The Adjacent<\/strong> side (along the x<\/mi>x<\/annotation><\/semantics><\/math>-axis) is 4<\/strong>.<\/li>\n\n\n\n
            • The Opposite<\/strong> side (parallel to the y<\/mi>y<\/annotation><\/semantics><\/math>-axis) is 3<\/strong>.<\/li>\n<\/ul>\n\n\n\n

              Step 3: Use the Tangent Ratio<\/h4>\n\n\n\n

              In trigonometry, tan<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b8<\/mi>)<\/mo>=<\/mo>Opposite<\/mtext>Adjacent<\/mtext><\/mfrac><\/mrow>\\tan(\\theta) = \\frac{\\text{Opposite}}{\\text{Adjacent}}<\/annotation><\/semantics><\/math>. For vectors, this is always:<\/p>\n\n\n\n

              tan<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b8<\/mi>)<\/mo>=<\/mo>y<\/mi>x<\/mi><\/mfrac><\/mrow>\\tan(\\theta) = \\frac{y}{x}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

              Substitute our values:<\/p>\n\n\n\n

              tan<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b8<\/mi>)<\/mo>=<\/mo>3<\/mn>4<\/mn><\/mfrac><\/mrow>\\tan(\\theta) = \\frac{3}{4}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

              Step 4: Solve for \u03b8<\/mi>\\theta<\/annotation><\/semantics><\/math><\/h4>\n\n\n\n

              To find the angle, use the inverse tangent function (tan<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>\u2061<\/mo><\/mspace><\/mrow>\\tan^{-1}<\/annotation><\/semantics><\/math>) on your calculator:<\/p>\n\n\n\n

              \u03b8<\/mi>=<\/mo>tan<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>\u2061<\/mo>(<\/mo>0.75<\/mn>)<\/mo><\/mrow>\\theta = \\tan^{-1}(0.75)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

              \u03b8<\/mi>\u2248<\/mo>36.87<\/mn>\u2218<\/mo><\/msup><\/mrow>\\theta \\approx 36.87^\\circ<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

              The Verdict:<\/strong> The direction of vector \ud835\udc2f<\/mi>\\mathbf{v}<\/annotation><\/semantics><\/math> is approximately 36.87<\/mn>\u2218<\/mo><\/msup>36.87^\\circ<\/annotation><\/semantics><\/math> from the positive x<\/mi>x<\/annotation><\/semantics><\/math>-axis.<\/p>\n\n\n\n
              \n\n\n\n

              \u26a0\ufe0f The “Year 11 Trap”: Watch your Quadrants!<\/h3>\n\n\n\n

              The formula \u03b8<\/mi>=<\/mo>tan<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>\u2061<\/mo>(<\/mo>y<\/mi>x<\/mi><\/mfrac>)<\/mo><\/mrow>\\theta = \\tan^{-1}(\\frac{y}{x})<\/annotation><\/semantics><\/math> works perfectly in the first quadrant. However, if your vector was \ud835\udc2e<\/mi>=<\/mo>\u2212<\/mo>4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{u} = -4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math> (Second Quadrant), your calculator might give you a negative number or a confusing result.<\/p>\n\n\n\n

              Queensland Tip:<\/strong> Always draw your sketch first.<\/p>\n\n\n\n

                \n
              • If your vector is in the 2nd Quadrant<\/strong>: Calculate the angle inside the triangle, then subtract it from 180<\/mn>\u2218<\/mo><\/msup>180^\\circ<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
              • If your vector is in the 3rd Quadrant<\/strong>: Add your angle to 180<\/mn>\u2218<\/mo><\/msup>180^\\circ<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
              • If your vector is in the 4th Quadrant<\/strong>: Subtract your angle from 360<\/mn>\u2218<\/mo><\/msup>360^\\circ<\/annotation><\/semantics><\/math> (or use a negative angle).<\/li>\n<\/ul>\n\n\n\n

                [Image showing vectors in all four quadrants and how to calculate their true angles from the positive x-axis]<\/p>\n\n\n\n

                \n\n\n\n

                Practice Challenge for you:<\/h3>\n\n\n\n

                If you have a vector \ud835\udc30<\/mi>=<\/mo>\u2212<\/mo>5<\/mn>\ud835\udc22<\/mi>\u2212<\/mo>5<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{w} = -5\\mathbf{i} – 5\\mathbf{j}<\/annotation><\/semantics><\/math>, it\u2019s pointing diagonally down into the third quadrant.<\/p>\n\n\n\n

                Can you guess what the angle from the positive <\/strong>x<\/mi>x<\/annotation><\/semantics><\/math>-axis would be before doing the math?<\/strong> (Hint: think about a 45<\/mn>\u2218<\/mo><\/msup>45^\\circ<\/annotation><\/semantics><\/math> angle added to 180<\/mn>\u2218<\/mo><\/msup>180^\\circ<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n

                **************************<\/p>\n\n\n\n

                Would you like me to show you how to convert these angles into Compass Bearings<\/strong> (like 045<\/mn>\u2218<\/mo><\/msup>045^\\circ<\/annotation><\/semantics><\/math> or N<\/mi>30<\/mn>\u2218<\/mo><\/msup>E<\/mi><\/mrow>N30^\\circ E<\/annotation><\/semantics><\/math>), which often pops up in Specialist Maths?<\/p>\n\n\n\n

                [Yes, please.]<\/p>\n\n\n\n

                In Queensland maths, especially when you get into the applications of vectors (like navigation or physics), you\u2019ll often be asked to give your answer as a Bearing<\/strong> instead of just an angle from the x<\/mi>x<\/annotation><\/semantics><\/math>-axis.<\/p>\n\n\n\n

                There are two main types you need to know: True Bearings<\/strong> and Compass (Quadrant) Bearings<\/strong>.<\/p>\n\n\n\n

                \n\n\n\n

                1. True Bearings (T<\/mi>T<\/annotation><\/semantics><\/math>)<\/h3>\n\n\n\n

                This is the standard used in aviation and shipping.<\/p>\n\n\n\n

                  \n
                • Starting Point:<\/strong> Always North (000<\/mn>\u2218<\/mo><\/msup>000^\\circ<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
                • Direction:<\/strong> Always clockwise<\/strong>.<\/li>\n\n\n\n
                • Format:<\/strong> Always written with three digits (e.g., 045<\/mn>\u2218<\/mo><\/msup>T<\/mi><\/mrow>045^\\circ T<\/annotation><\/semantics><\/math>).<\/li>\n<\/ul>\n\n\n\n

                  The Conversion Secret:<\/strong><\/p>\n\n\n\n

                  On your Cartesian plane, “North” is the positive y<\/mi>y<\/annotation><\/semantics><\/math>-axis (90<\/mn>\u2218<\/mo><\/msup>90^\\circ<\/annotation><\/semantics><\/math>). If your math angle is \u03b8<\/mi>\\theta<\/annotation><\/semantics><\/math>, the True Bearing is usually 90<\/mn>\u2218<\/mo><\/msup>\u2212<\/mo>\u03b8<\/mi><\/mrow>90^\\circ – \\theta<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

                  [Image comparing Cartesian angles (anti-clockwise from East) vs True Bearings (clockwise from North)]<\/p>\n\n\n\n

                  \n\n\n\n

                  2. Compass (Quadrant) Bearings<\/h3>\n\n\n\n

                  These describe the direction by starting at North or South and “turning” towards East or West.<\/p>\n\n\n\n

                    \n
                  • Format:<\/strong> [Start Direction] [Angle] [Turn Direction]<\/li>\n\n\n\n
                  • Examples:<\/strong> N<\/mi>30<\/mn>\u2218<\/mo><\/msup>E<\/mi><\/mrow>N 30^\\circ E<\/annotation><\/semantics><\/math> (Start North, turn (Start South, turn 20<\/mn>\u2218<\/mo><\/msup>20^\\circ<\/annotation><\/semantics><\/math> West).<\/li>\n<\/ul>\n\n\n\n
                    \n\n\n\n

                    Worked Example: Converting our Vector<\/h3>\n\n\n\n

                    Let\u2019s use our vector from before: \ud835\udc2f<\/mi>=<\/mo>4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{v} = 4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

                    We found the mathematical angle was 36.87<\/mn>\u2218<\/mo><\/msup>36.87^\\circ<\/annotation><\/semantics><\/math> (measured from the x<\/mi>x<\/annotation><\/semantics><\/math>-axis\/East).<\/p>\n\n\n\n

                    To find the True Bearing:<\/h4>\n\n\n\n
                      \n
                    1. North is 90<\/mn>\u2218<\/mo><\/msup>90^\\circ<\/annotation><\/semantics><\/math> on our grid.<\/li>\n\n\n\n
                    2. East is 0<\/mn>\u2218<\/mo><\/msup>0^\\circ<\/annotation><\/semantics><\/math> on our grid.<\/li>\n\n\n\n
                    3. Our vector is 36.87<\/mn>\u2218<\/mo><\/msup>36.87^\\circ<\/annotation><\/semantics><\/math> “up” from East.<\/li>\n\n\n\n
                    4. The gap between North and our vector is: 90<\/mn>\u2218<\/mo><\/msup>\u2212<\/mo>36.87<\/mn>\u2218<\/mo><\/msup>=<\/mo>53.13<\/mn>\u2218<\/mo><\/msup><\/mrow>90^\\circ – 36.87^\\circ = 53.13^\\circ<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
                    5. True Bearing:<\/strong> 053.13<\/mn>\u2218<\/mo><\/msup>T<\/mi><\/mrow>053.13^\\circ T<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n\n\n\n

                      To find the Compass Bearing:<\/h4>\n\n\n\n
                        \n
                      1. Look at the diagram. Is the vector closer to North or South? North<\/strong>.<\/li>\n\n\n\n
                      2. Is it leaning towards East or West? East<\/strong>.<\/li>\n\n\n\n
                      3. The angle from North toward East is 53.13<\/mn>\u2218<\/mo><\/msup>53.13^\\circ<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
                      4. Compass Bearing:<\/strong> N<\/mi>53.13<\/mn>\u2218<\/mo><\/msup>E<\/mi><\/mrow>N 53.13^\\circ E<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n\n\n\n

                        [Image showing how a vector in the first quadrant is converted into both True and Compass bearings]<\/p>\n\n\n\n

                        \n\n\n\n

                        Summary Table for the 4 Quadrants<\/h3>\n\n\n\n
                        Quadrant<\/strong><\/td>Vector Example<\/strong><\/td>Cartesian Angle (\u03b8)<\/strong><\/td>True Bearing (T)<\/strong><\/td><\/tr><\/thead>
                        1st (NE)<\/strong>
                        <\/td>                        		4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math><\/td>36.9<\/mn>\u2218<\/mo><\/msup>36.9^\\circ<\/annotation><\/semantics><\/math><\/td>90<\/mn>\u2212<\/mo>36.9<\/mn><\/mrow>90 – 36.9 <\/annotation><\/semantics><\/math>
                        =<\/mo>\ud835\udfce\ud835\udfd3\ud835\udfd1.\ud835\udfcf<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow><\/mrow>= \\mathbf{053.1^\\circ T}<\/annotation><\/semantics><\/math><\/td><\/tr>
                        2nd (NW)<\/strong><\/td>\u2212<\/mo>4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>-4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math><\/td>143.1<\/mn>\u2218<\/mo><\/msup>143.1^\\circ<\/annotation><\/semantics><\/math><\/td>450<\/mn>\u2212<\/mo>143.1<\/mn><\/mrow>450 – 143.1<\/annotation><\/semantics><\/math> = \ud835\udfd1\ud835\udfce\ud835\udfd4.\ud835\udfd7<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow>\\mathbf{306.9^\\circ T}<\/annotation><\/semantics><\/math>*<\/td><\/tr>
                        3rd (SW)<\/strong><\/td>\u2212<\/mo>4<\/mn>\ud835\udc22<\/mi>\u2212<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>-4\\mathbf{i} – 3\\mathbf{j}<\/annotation><\/semantics><\/math><\/td>216.9<\/mn>\u2218<\/mo><\/msup>216.9^\\circ<\/annotation><\/semantics><\/math><\/td>450<\/mn>\u2212<\/mo>216.9<\/mn>=<\/mo><\/mrow>450 – 216.9 =<\/annotation><\/semantics><\/math> \ud835\udfd0\ud835\udfd1\ud835\udfd1.\ud835\udfcf<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow>\\mathbf{233.1^\\circ T}<\/annotation><\/semantics><\/math><\/td><\/tr>
                        4th (SE)<\/strong><\/td>4<\/mn>\ud835\udc22<\/mi>\u2212<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>4\\mathbf{i} – 3\\mathbf{j}<\/annotation><\/semantics><\/math><\/td>323.1<\/mn>\u2218<\/mo><\/msup>323.1^\\circ<\/annotation><\/semantics><\/math><\/td>450<\/mn>\u2212<\/mo>323.1<\/mn>=<\/mo><\/mrow>450 – 323.1 = <\/annotation><\/semantics><\/math>\ud835\udfcf\ud835\udfd0\ud835\udfd4.\ud835\udfd7<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow>\\mathbf{126.9^\\circ T}<\/annotation><\/semantics><\/math><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

                        *Note: A quick hack for True Bearings is <\/em>T<\/mi>r<\/mi>u<\/mi>e<\/mi>=<\/mo>(<\/mo>450<\/mn>\u2212<\/mo>\u03b8<\/mi>)<\/mo><\/mo><\/mspace>(<\/mo><\/mtext>mod<\/mi><\/mrow><\/mspace>360<\/mn>)<\/mo><\/mrow>True = (450 – \\theta) \\pmod{360}<\/annotation><\/semantics><\/math>. But honestly? Just drawing the picture and looking at the “North” line is much safer!<\/em><\/p>\n\n\n\n
                        \n\n\n\n

                        Pro-Tip for Exams<\/h3>\n\n\n\n

                        If a question mentions “a bearing of…”<\/strong>, change your brain to “Clockwise from North” mode immediately. Many students lose marks because they calculate the correct angle but from the wrong starting line.<\/p>\n\n\n\n

                        Would you like to try converting a “Third Quadrant” vector (where both <\/strong>x<\/mi>x<\/annotation><\/semantics><\/math> and <\/strong>y<\/mi>y<\/annotation><\/semantics><\/math> are negative) to see if you can nail the bearing?<\/strong><\/p>\n\n\n\n

                        ************************************************************************************************<\/p>\n\n\n\n

                        I11\u5e74\u751f\u3078\u3088\u3046\u3053\u305d\uff01\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e\u5c02\u9580\u6570\u5b66\u3084\u6570\u5b66\u7684\u624b\u6cd5\uff08QCAA\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\uff09\u3092\u5b66\u307c\u3046\u3068\u3057\u3066\u3044\u308b\u306a\u3089\u3001\u30d9\u30af\u30c8\u30eb\u306f\u3042\u306a\u305f\u306e\u65b0\u3057\u3044\u89aa\u53cb\u306b\u306a\u308b\u3067\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

                        \u30d9\u30af\u30c8\u30eb\u3092\u6570\u5b66\u7684\u306a\u6307\u793a<\/strong>\u3068\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u3053\u308c\u307e\u3067\u306f\u4e3b\u306b\u300c\u30b9\u30ab\u30e9\u30fc\u300d\u3001\u3064\u307e\u308a\u91cf<\/em>\uff08\u4f8b\u3048\u30705kg\u306e\u30ea\u30f3\u30b4\u308425\u2103\u306e\u6e29\u5ea6\u306a\u3069\uff09\u3060\u3051\u3092\u793a\u3059\u6570\u5024\u3092\u6271\u3063\u3066\u304d\u307e\u3057\u305f\u3002\u30d9\u30af\u30c8\u30eb\u306f\u3001\u8ddd\u96e2<\/strong>\u3068\u65b9\u5411<\/strong>\u306e2\u3064\u306e\u3053\u3068\u3092\u540c\u6642\u306b\u793a\u3057\u3066\u304f\u308c\u308b\u306e\u3067\u3001\u3088\u308a\u5206\u304b\u308a\u3084\u3059\u3044\u3067\u3059\u3002<\/p>\n\n\n\n

                        \n\n\n\n

                        1. \u57fa\u672c\uff1a\u5927\u304d\u3055\u3068\u65b9\u5411<\/h4>\n\n\n\n

                        \u76f4\u4ea4\u5e73\u9762\uff08\u6a19\u6e96\u5ea7\u6a19\uff58<\/mtext>\uff58<\/annotation><\/semantics><\/math>\u3068y<\/mi>y<\/annotation><\/semantics><\/math>\u30b0\u30ea\u30c3\u30c9\uff09\u3067\u306f\u3001\u30d9\u30af\u30c8\u30eb\u306f\u77e2\u5370\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n
                          \n
                        • \u5927\u304d\u3055:<\/strong> \u77e2\u5370\u306e\u9577\u3055\uff08\u8ddd\u96e2\uff09\u3067\u3059\u3002<\/li>\n\n\n\n
                        • \u65b9\u5411:<\/strong> \u77e2\u5370\u306e\u5411\u304d\u3067\u3059\u3002<\/li>\n<\/ul>\n\n\n\n

                          \u5b66\u6821\u306e\u9580\u304b\u3089\u58f2\u5e97\u307e\u3067\u306e\u884c\u304d\u65b9\u3092\u53cb\u9054\u306b\u6559\u3048\u3066\u3044\u308b\u3068\u60f3\u50cf\u3057\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002\u300c50\u30e1\u30fc\u30c8\u30eb\u6b69\u304f\u300d\u3068\u306f\u8a00\u3044\u307e\u305b\u3093\u3002\u300c\u6771<\/strong>\u306b50\u30e1\u30fc\u30c8\u30eb\u6b69\u304f\u300d\u3068\u8a00\u3046\u3067\u3057\u3087\u3046\u3002\u3053\u308c\u304c\u30d9\u30af\u30c8\u30eb\u3067\u3059\u3002<\/p>\n\n\n\n

                          \n\n\n\n

                          2. \u30d9\u30af\u30c8\u30eb\u306e\u66f8\u304d\u65b9\uff08\u8868\u8a18\u6cd5\uff09<\/h4>\n\n\n\n

                          \u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e\u6559\u5e2b\u306f\u6b63\u3057\u3044\u8868\u8a18\u6cd5\u3092\u91cd\u8996\u3057\u3066\u3044\u307e\u3059\u3002\u30d9\u30af\u30c8\u30eb\u306f\u3044\u304f\u3064\u304b\u306e\u65b9\u6cd5\u3067\u8868\u8a18\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

                            \n
                          • \u592a\u5b57\uff1a<\/strong> \ud835\udc2e<\/mi>\\mathbf{u}<\/annotation><\/semantics><\/math><\/strong>\u307e\u305f\u306f\ud835\udc2f<\/mi>\\mathbf{v}<\/annotation><\/semantics><\/math><\/strong>\uff08\u6559\u79d1\u66f8\u3067\u3088\u304f\u4f7f\u308f\u308c\u307e\u3059\uff09\u3002<\/li>\n\n\n\n
                          • \u77e2\u5370\u8868\u8a18\uff1a<\/strong>A<\/mi>B<\/mi><\/mrow>\u2192<\/mo><\/mover>\\vec{AB}<\/annotation><\/semantics><\/math>\uff08A<\/mi>A<\/annotation><\/semantics><\/math>\u70b9\u304b\u3089B<\/mi>B<\/annotation><\/semantics><\/math>\u70b9\u3078\u306e\u7d4c\u8def\u3092\u610f\u5473\u3057\u307e\u3059\uff09\u3002<\/li>\n\n\n\n
                          • \u5217\u30d9\u30af\u30c8\u30eb\uff1a<\/strong> [<\/mo>x<\/mi><\/mtd><\/mtr>y<\/mi><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>\\begin{bmatrix} x \\\\ y \\end{bmatrix}<\/annotation><\/semantics><\/math>\u3002\u3053\u308c\u306fGPS\u5ea7\u6a19\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3059\u3002\u4e0a\u306e\u6570\u5b57\u306f\u6c34\u5e73\u65b9\u5411\u306b\u4f55\u5358\u4f4d\u79fb\u52d5\u3059\u308b\u304b\u3092\u793a\u3057\u3001\u4e0b\u306e\u6570\u5b57\u306f\u5782\u76f4\u65b9\u5411\u306b\u4f55\u5358\u4f4d\u79fb\u52d5\u3059\u308b\u304b\u3092\u793a\u3057\u307e\u3059\u3002<\/li>\n\n\n\n
                          • \u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u5f62\u5f0f\uff1a<\/strong>x<\/mi>\ud835\udc22<\/mi>+<\/mo>y<\/mi>\ud835\udc23<\/mi><\/mrow>x\\mathbf{i} + y\\mathbf{j}<\/annotation><\/semantics><\/math> \u3002<\/li>\n\n\n\n
                          • \ud835\udc22<\/mi>\\mathbf{i}<\/annotation><\/semantics><\/math>\u306fx<\/mi>x<\/annotation><\/semantics><\/math>\u65b9\u5411\u3078\u306e1\u6b69\u3067\u3059\u3002<\/li>\n\n\n\n
                          • \ud835\udc23<\/mi>\\mathbf{j}<\/annotation><\/semantics><\/math>\u306fy<\/mi>y<\/annotation><\/semantics><\/math>\u65b9\u5411\u3078\u306e1\u6b69\u3067\u3059\u3002<\/li>\n<\/ul>\n\n\n\n
                            \n

                            \u4f8b\uff1a<\/strong>\ud835\udc2f<\/mi>=<\/mo>3<\/mn>\ud835\udc22<\/mi>+<\/mo>2<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{v} = 3\\mathbf{i} + 2\\mathbf{j}<\/annotation><\/semantics><\/math> \u306e\u5834\u5408\u3001\u53f3\u306b3\u6b69\u3001\u4e0a\u306b2\u6b69\u9032\u307f\u307e\u3059\u3002<\/p>\n<\/blockquote>\n\n\n\n
                            \n\n\n\n

                            3. \u30d9\u30af\u30c8\u30eb\u306e\u300c\u6570\u5b66\u300d\uff08\u898b\u305f\u76ee\u3088\u308a\u7c21\u5358\u3067\u3059\uff09<\/h4>\n\n\n\n

                            \u30d9\u30af\u30c8\u30eb\u3092\u4f7f\u3063\u3066\u7b97\u8853\u6f14\u7b97\u3092\u884c\u3046\u3053\u3068\u306f\u3067\u304d\u307e\u3059\u304c\u3001\u901a\u5e38\u306e\u4ee3\u6570\u3088\u308a\u3082\u8996\u899a\u7684\u306b\u7406\u89e3\u3057\u3084\u3059\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

                            \u30d9\u30af\u30c8\u30eb\u306e\u52a0\u7b97\uff08\u300c\u5148\u7aef\u304b\u3089\u672b\u5c3e\u307e\u3067\u300d\u306e\u30eb\u30fc\u30eb\uff09<\/h4>\n\n\n\n

                            \u30d9\u30af\u30c8\u30eb\ud835\udc1a<\/mi>\\mathbf{a}<\/annotation><\/semantics><\/math>\u3068\u30d9\u30af\u30c8\u30eb \ud835\udc1b<\/mi>\\mathbf{b}<\/annotation><\/semantics><\/math>\u3092\u52a0\u7b97\u3059\u308b\u306b\u306f\u3001\ud835\udc1b<\/mi>\\mathbf{b}<\/annotation><\/semantics><\/math> \u306e\u300c\u672b\u5c3e\u300d\u3092 \ud835\udc1a<\/mi>\\mathbf{a}<\/annotation><\/semantics><\/math>\u306e\u300c\u5148\u7aef\u300d\u306b\u7f6e\u304d\u307e\u3059\u3002\u7d50\u679c\uff08\u7d50\u679c\u30d9\u30af\u30c8\u30eb<\/strong>\uff09\u306f\u3001\ud835\udc1a<\/mi>\\mathbf{a}<\/annotation><\/semantics><\/math>\u306e\u6700\u521d\u304b\u3089\ud835\udc1b<\/mi>\\mathbf{b}<\/annotation><\/semantics><\/math>\u306e\u6700\u5f8c\u307e\u3067\u306e\u8fd1\u9053\u3067\u3059\u3002<\/p>\n\n\n\n

                            \u30b9\u30ab\u30e9\u30fc\u4e57\u7b97<\/h4>\n\n\n\n

                            \u30d9\u30af\u30c8\u30eb\u306b\u901a\u5e38\u306e\u6570\uff08\u30b9\u30ab\u30e9\u30fc\uff09\u3092\u4e57\u7b97\u3059\u308b\u5834\u5408\u306f\u3001\u30d9\u30af\u30c8\u30eb\u306e\u9577\u3055\u3092\u5909\u3048\u308b\u3060\u3051\u3067\u3059\u3002<\/p>\n\n\n\n

                              \n
                            • 2<\/mn>\ud835\udc2f<\/mi><\/mrow>2\\mathbf{v}<\/annotation><\/semantics><\/math>\u306f \ud835\udc2f<\/mi>\\mathbf{v}<\/annotation><\/semantics><\/math>\u306e2\u500d\u306e\u9577\u3055\u3067\u3001\u540c\u3058\u65b9\u5411\u3092\u5411\u3044\u3066\u3044\u307e\u3059\u3002<\/li>\n\n\n\n
                            • \u2212<\/mo>1<\/mn>\ud835\udc2f<\/mi><\/mrow>-1\\mathbf{v}<\/annotation><\/semantics><\/math>\u306f\u540c\u3058\u9577\u3055\u3067\u3059\u304c\u3001\u6b63\u53cd\u5bfe\u306e\u65b9\u5411<\/strong>\u3092\u5411\u3044\u3066\u3044\u307e\u3059\u3002<\/li>\n<\/ul>\n\n\n\n
                              \n\n\n\n

                              4. \u5927\u304d\u3055\uff08\u5927\u304d\u3055\uff09\u306e\u8a08\u7b97<\/h4>\n\n\n\n

                              \u5e73\u9762\u4e0a\u306e\u30d9\u30af\u30c8\u30eb\u306f\u3001\uff58<\/mtext>\uff58<\/annotation><\/semantics><\/math>\u8ef8\u3068\uff59<\/mtext>\uff59<\/annotation><\/semantics><\/math>\u8ef8\u3092\u6301\u3064\u76f4\u89d2\u4e09\u89d2\u5f62\u3092\u5f62\u6210\u3059\u308b\u305f\u3081\u3001\u9577\u3055\u3092\u6c42\u3081\u308b\u306b\u306f\u304a\u306a\u3058\u307f\u306e\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406<\/strong>\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

                              \u30d9\u30af\u30c8\u30eb \ud835\udc2f<\/mi>=<\/mo>[<\/mo>x<\/mi><\/mtd><\/mtr>y<\/mi><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mrow>\\mathbf{v} = \\begin{bmatrix} x \\\\ y \\end{bmatrix}<\/annotation><\/semantics><\/math>\u306e\u5834\u5408\u3001\u5927\u304d\u3055\uff08 |<\/mi>\ud835\udc2f<\/mi>|<\/mi><\/mrow>|\\mathbf{v}|<\/annotation><\/semantics><\/math>\u3068\u8868\u8a18\uff09\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
                              |<\/mi>v<\/mtext>|<\/mi><\/mrow>=<\/mo>x<\/mi>2<\/mn><\/msup>+<\/mo>y<\/mi>2<\/mn><\/msup><\/mrow><\/msqrt><\/mrow>\\mathbf{|\\text{v}|} = \\sqrt{x^2 + y^2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
                              \n\n\n\n

                              5. \u306a\u305c\u5fc5\u8981\u306a\u306e\u304b\uff1f<\/h4>\n\n\n\n

                              \u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\u3067\u306f\u3001\u3053\u308c\u3089\u306e\u5b9a\u7406\u3092\u7528\u3044\u3066\u3001\u6b21\u306e\u3088\u3046\u306a\u73fe\u5b9f\u4e16\u754c\u306e\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

                                \n
                              1. \u901f\u5ea6:<\/strong> \u5317\u3078500km\/h\u3067\u98db\u884c\u3059\u308b\u98db\u884c\u6a5f\u3068\u3001\u6771\u307850km\/h\u3067\u5439\u304f\u98a8\u3002<\/li>\n\n\n\n
                              2. \u529b:<\/strong> 2\u4eba\u304c\u7570\u306a\u308b\u65b9\u5411\u306b\u30ed\u30fc\u30d7\u3092\u5f15\u304f\u3002<\/li>\n\n\n\n
                              3. \u5e7e\u4f55\u5b66:<\/strong> \u4e2d\u70b9\u3068\u5bfe\u89d2\u7dda\u3092\u7528\u3044\u3066\u56f3\u5f62\u306b\u95a2\u3059\u308b\u4e8b\u67c4\u3092\u8a3c\u660e\u3059\u308b\u3002<\/li>\n<\/ol>\n\n\n\n

                                \u89aa\u5207\u306aAI\u304b\u3089\u306e\u30a2\u30c9\u30d0\u30a4\u30b9:<\/strong> \u5fc5\u305a\u56f3\u3092\u63cf\u304d\u307e\u3057\u3087\u3046\u3002\u305f\u3068\u3048\u982d\u306e\u4e2d\u3067\u7b54\u3048\u304c\u5206\u304b\u3063\u3066\u3044\u308b\u3068\u601d\u3063\u3066\u3082\u3001\u77e2\u5370\u3092\u63cf\u304f\u3053\u3068\u3067\u3001\u9ad8\u68213\u5e74\u751f\u306e\u8a66\u9a13\u3067\u3088\u304f\u3042\u308b\u300c\u304a\u304b\u3057\u306a\u300d\u65b9\u5411\u306e\u9593\u9055\u3044\u3092\u9632\u3050\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

                                \n\n\n\n

                                ****************************************************************************<\/p>\n\n\n\n

                                \u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\u3067\u306f\u3001\u30d9\u30af\u30c8\u30eb\u306e\u300c\u65b9\u5411\u300d\u3092\u6c42\u3081\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u3001\u901a\u5e38\u3001\u6b63\u306e<\/strong>x<\/mi>x<\/annotation><\/semantics><\/math>-\u8ef8<\/strong>\uff08\u53cd\u6642\u8a08\u56de\u308a\uff09\u3068\u306e\u89d2\u5ea6\u03b8<\/mi>\\theta<\/annotation><\/semantics><\/math>\u3092\u6c42\u3081\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u3001\u3068\u304d\u306b\u306f\u65b9\u4f4d<\/strong>\u3067\u8868\u3055\u308c\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

                                \u5178\u578b\u7684\u306a\u8a66\u9a13\u5f62\u5f0f\u306e\u554f\u984c\u3092\u898b\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

                                \n\n\n\n

                                \u554f\u984c<\/h4>\n\n\n\n

                                \u30d9\u30af\u30c8\u30eb \ud835\udc2f<\/mi>=<\/mo>4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{v} = 4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math> \u306e\u65b9\u5411\u3092\u6c42\u3081\u306a\u3055\u3044\u3002<\/p>\n\n\n\n

                                \u30b9\u30c6\u30c3\u30d71\uff1a\u30d9\u30af\u30c8\u30eb\u3092\u30b9\u30b1\u30c3\u30c1\u3059\u308b<\/h4>\n\n\n\n

                                \u96fb\u5353\u3092\u4f7f\u3046\u524d\u306b\u3001\u30d9\u30af\u30c8\u30eb\u3092\u8996\u899a\u5316\u3057\u307e\u3057\u3087\u3046\u3002\u4e21\u65b9\u306e\u8981\u7d20\u304c\u6b63\u306a\u306e\u3067\u3001\u30d9\u30af\u30c8\u30eb\u306f\u53f3\u306b4\u5358\u4f4d<\/strong>\u3001\u4e0a\u306b3\u5358\u4f4d<\/strong>\u79fb\u52d5\u3057\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u7b2c1\u8c61\u9650<\/strong>\u306b\u5165\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

                                \u30b9\u30c6\u30c3\u30d72\uff1a\u4e09\u89d2\u5f62\u3092\u7279\u5b9a\u3059\u308b<\/h4>\n\n\n\n

                                \u30d9\u30af\u30c8\u30eb\u306e\u5148\u7aef\u304b\u3089-\u8ef8\u307e\u3067\u7dda\u3092\u5f15\u304f\u3068\u3001\u4ee5\u4e0b\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u76f4\u89d2\u4e09\u89d2\u5f62\u304c\u4f5c\u6210\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

                                  \n
                                • \u96a3\u63a5<\/strong>\u8fba\uff08x<\/mi>x<\/annotation><\/semantics><\/math>-\u8ef8\u306b\u6cbf\u3063\u305f\uff09\u306f4<\/strong>\u3067\u3059\u3002<\/li>\n\n\n\n
                                • \u5bfe\u8fba<\/strong>\uff08y<\/mi>y<\/annotation><\/semantics><\/math>-\u8ef8\u306b\u5e73\u884c\u306a\u8fba\uff09\u306f3<\/strong>\u3067\u3059\u3002<\/li>\n<\/ul>\n\n\n\n

                                  \u30b9\u30c6\u30c3\u30d73\uff1a\u6b63\u63a5\u6bd4\u3092\u4f7f\u3046<\/h4>\n\n\n\n

                                  \u4e09\u89d2\u6cd5\u3067\u306f\u3001tan<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b8<\/mi>)<\/mo>=<\/mo>Opposite<\/mtext>Adjacent<\/mtext><\/mfrac><\/mrow>\\tan(\\theta) = \\frac{\\text{Opposite}}{\\text{Adjacent}}<\/annotation><\/semantics><\/math> \u3067\u3059\u3002\u30d9\u30af\u30c8\u30eb\u306e\u5834\u5408\u3001\u3053\u308c\u306f\u5e38\u306b\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002tan<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b8<\/mi>)<\/mo>=<\/mo>y<\/mi>x<\/mi><\/mfrac><\/mrow>\\tan(\\theta) = \\frac{y}{x}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                                  \u5024\u3092\u4ee3\u5165\u3057\u307e\u3059\u3002tan<\/mi>\u2061<\/mo><\/mrow>(<\/mo>\u03b8<\/mi>)<\/mo>=<\/mo>3<\/mn>4<\/mn><\/mfrac><\/mrow>\\tan(\\theta) = \\frac{3}{4}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                                  \u30b9\u30c6\u30c3\u30d74\uff1a \u03b8<\/mi>\\theta<\/annotation><\/semantics><\/math>\u3092\u89e3\u304d\u307e\u3059\u3002<\/h4>\n\n\n\n

                                  \u89d2\u5ea6\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\u96fb\u5353\u306e\u9006\u6b63\u63a5\u95a2\u6570\uff08tan<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>\u2061<\/mo><\/mspace><\/mrow>\\tan^{-1}<\/annotation><\/semantics><\/math>\uff09\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

                                  \u03b8<\/mi>=<\/mo>tan<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>\u2061<\/mo>(<\/mo>0.75<\/mn>)<\/mo><\/mrow>\\theta = \\tan^{-1}(0.75)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                                  \u03b8<\/mi>\u2248<\/mo>36.87<\/mn>\u2218<\/mo><\/msup><\/mrow>\\theta \\approx 36.87^\\circ<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                                  \u7d50\u8ad6\uff1a<\/strong> \u30d9\u30af\u30c8\u30eb\u306e\u65b9\u5411\u306f\u3001\u6b63\u306ex<\/mi>x<\/annotation><\/semantics><\/math>-\u8ef8\u304b\u3089\u7d0436.87<\/mn>\u2218<\/mo><\/msup>36.87^\\circ<\/annotation><\/semantics><\/math>\u3067\u3059\u3002<\/p>\n\n\n\n
                                  \n\n\n\n

                                  \u26a0\ufe0f\u300c\u9ad8\u68213\u5e74\u751f\u306e\u843d\u3068\u3057\u7a74\u300d\uff1a\u8c61\u9650\u306b\u6ce8\u610f\uff01<\/h3>\n\n\n\n

                                  \u3053\u306e\u516c\u5f0f \u03b8<\/mi>=<\/mo>tan<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>\u2061<\/mo>(<\/mo>y<\/mi>x<\/mi><\/mfrac>)<\/mo><\/mrow>\\theta = \\tan^{-1}(\\frac{y}{x})<\/annotation><\/semantics><\/math> \u306f\u7b2c1\u8c61\u9650\u3067\u306f\u5b8c\u74a7\u306b\u6a5f\u80fd\u3057\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u30d9\u30af\u30c8\u30eb\u304c \ud835\udc2e<\/mi>=<\/mo>\u2212<\/mo>4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{u} = -4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math>\uff08\u7b2c2\u8c61\u9650\uff09\u306e\u5834\u5408\u3001\u96fb\u5353\u306f\u8ca0\u306e\u5024\u3084\u6df7\u4e71\u3092\u62db\u304f\u7d50\u679c\u3092\u8868\u793a\u3059\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

                                  \u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u3067\u306e\u30d2\u30f3\u30c8\uff1a<\/strong> \u5fc5\u305a\u6700\u521d\u306b\u30b9\u30b1\u30c3\u30c1\u3092\u63cf\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

                                    \n
                                  • \u30d9\u30af\u30c8\u30eb\u304c\u7b2c2\u8c61\u9650<\/strong>\u306b\u3042\u308b\u5834\u5408\uff1a\u4e09\u89d2\u5f62\u306e\u5185\u5074\u306e\u89d2\u5ea6\u3092\u8a08\u7b97\u3057\u3001\u305d\u308c\u3092 180<\/mn>\u2218<\/mo><\/msup>180^\\circ<\/annotation><\/semantics><\/math>\u304b\u3089\u5f15\u304d\u307e\u3059\u3002<\/li>\n\n\n\n
                                  • \u30d9\u30af\u30c8\u30eb\u304c\u7b2c3\u8c61\u9650<\/strong>\u306b\u3042\u308b\u5834\u5408\uff1a\u89d2\u5ea6\u3092 180<\/mn>\u2218<\/mo><\/msup>180^\\circ<\/annotation><\/semantics><\/math>\u306b\u52a0\u3048\u307e\u3059\u3002<\/li>\n\n\n\n
                                  • \u30d9\u30af\u30c8\u30eb\u304c\u7b2c4\u8c61\u9650<\/strong>\u306b\u3042\u308b\u5834\u5408\uff1a\u89d2\u5ea6\u3092 360<\/mn>\u2218<\/mo><\/msup>360^\\circ<\/annotation><\/semantics><\/math> \u304b\u3089\u5f15\u304d\u307e\u3059\uff08\u307e\u305f\u306f\u8ca0\u306e\u89d2\u5ea6\u3092\u4f7f\u7528\u3057\u307e\u3059\uff09\u3002<\/li>\n<\/ul>\n\n\n\n

                                    [4\u3064\u306e\u8c61\u9650\u3059\u3079\u3066\u306b\u3042\u308b\u30d9\u30af\u30c8\u30eb\u3068\u3001\u305d\u308c\u3089\u306e\u6b63\u306ex\u8ef8\u304b\u3089\u306e\u771f\u306e\u89d2\u5ea6\u306e\u8a08\u7b97\u65b9\u6cd5\u3092\u793a\u3059\u753b\u50cf]<\/p>\n\n\n\n

                                    \n\n\n\n

                                    \u7df4\u7fd2\u554f\u984c\uff1a<\/h3>\n\n\n\n

                                    \u30d9\u30af\u30c8\u30eb \ud835\udc30<\/mi>=<\/mo>\u2212<\/mo>5<\/mn>\ud835\udc22<\/mi>\u2212<\/mo>5<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{w} = -5\\mathbf{i} – 5\\mathbf{j}<\/annotation><\/semantics><\/math> \u304c\u3042\u308b\u5834\u5408\u3001\u305d\u308c\u306f\u659c\u3081\u4e0b\u5411\u304d\u306b\u7b2c3\u8c61\u9650\u306b\u5411\u304b\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

                                    \u8a08\u7b97\u3092\u59cb\u3081\u308b\u524d\u306b\u3001\u6b63\u306ex\u8ef8\u304b\u3089\u306e\u89d2\u5ea6\u304c\u3069\u308c\u304f\u3089\u3044\u306b\u306a\u308b\u304b\u63a8\u6e2c\u3067\u304d\u307e\u3059\u304b\uff1f<\/strong> (\u30d2\u30f3\u30c8\uff1a180<\/mn>\u2218<\/mo><\/msup>180^\\circ<\/annotation><\/semantics><\/math> \u306b\u89d2\u5ea6 45<\/mn>\u2218<\/mo><\/msup>45^\\circ<\/annotation><\/semantics><\/math> \u3092\u8db3\u3059\u3053\u3068\u3092\u8003\u3048\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002)<\/p>\n\n\n\n

                                    \u3053\u308c\u3089\u306e\u89d2\u5ea6\u3092\u3001\u5c02\u9580\u6570\u5b66\u3067\u3088\u304f\u51fa\u3066\u304f\u308b\u30b3\u30f3\u30d1\u30b9\u65b9\u4f4d<\/strong>\uff08 045<\/mn>\u2218<\/mo><\/msup>045^\\circ<\/annotation><\/semantics><\/math>\u3084 N<\/mi>30<\/mn>\u2218<\/mo><\/msup>E<\/mi><\/mrow>N30^\\circ E<\/annotation><\/semantics><\/math>\u306a\u3069\uff09\u306b\u5909\u63db\u3059\u308b\u65b9\u6cd5\u3092\u304a\u6559\u3048\u3057\u307e\u3057\u3087\u3046\u304b\uff1f<\/p>\n\n\n\n

                                    **********************<\/p>\n\n\n\n

                                    \u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e\u6570\u5b66\u3001\u7279\u306b\u30d9\u30af\u30c8\u30eb\u306e\u5fdc\u7528\uff08\u822a\u6d77\u3084\u7269\u7406\u5b66\u306a\u3069\uff09\u3067\u306f\u3001x<\/mi>x<\/annotation><\/semantics><\/math>-\u8ef8\u304b\u3089\u306e\u89d2\u5ea6\u3060\u3051\u3067\u306a\u304f\u3001\u65b9\u4f4d<\/strong>\u3067\u7b54\u3048\u3092\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u304c\u3088\u304f\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

                                    \u77e5\u3063\u3066\u304a\u304f\u3079\u304d\u4e3b\u306a\u65b9\u4f4d\u306e\u7a2e\u985e\u306f2\u3064\u3042\u308a\u307e\u3059\u3002\u771f\u65b9\u4f4d<\/strong>\u3068\u30b3\u30f3\u30d1\u30b9\uff08\u56db\u5206\u5100\uff09\u65b9\u4f4d<\/strong>\u3067\u3059\u3002<\/p>\n\n\n\n

                                    \n\n\n\n

                                    1. \u771f\u65b9\u4f4d (T<\/mi>T<\/annotation><\/semantics><\/math>)<\/h3>\n\n\n\n

                                    \u3053\u308c\u306f\u822a\u7a7a\u3068\u8239\u8236\u3067\u4f7f\u7528\u3055\u308c\u308b\u6a19\u6e96\u3067\u3059\u3002<\/p>\n\n\n\n

                                      \n
                                    • \u51fa\u767a\u70b9:<\/strong> \u5e38\u306b\u5317 (000<\/mn>\u2218<\/mo><\/msup>000^\\circ<\/annotation><\/semantics><\/math>)\u3002<\/li>\n\n\n\n
                                    • \u65b9\u5411:<\/strong> \u5e38\u306b \u6642\u8a08\u56de\u308a<\/strong>\u3002<\/li>\n\n\n\n
                                    • \u66f8\u5f0f:<\/strong> \u5e38\u306b3\u6841\u3067\u8868\u8a18\u3057\u307e\u3059\uff08\u4f8b: 045<\/mn>\u2218<\/mo><\/msup>T<\/mi><\/mrow>045^\\circ T<\/annotation><\/semantics><\/math>\uff09\u3002<\/li>\n<\/ul>\n\n\n\n

                                      \u5909\u63db\u306e\u79d8\u8a23:<\/strong>
                                      \u76f4\u4ea4\u5ea7\u6a19\u5e73\u9762\u3067\u306f\u3001\u300c\u5317\u300d\u306fy<\/mi>y<\/annotation><\/semantics><\/math>-\u8ef8\u306e\u6b63\u65b9\u5411 (90<\/mn>\u2218<\/mo><\/msup>90^\\circ<\/annotation><\/semantics><\/math>) \u3067\u3059\u3002\u6570\u5b66\u7684\u306a\u89d2\u5ea6\u304c \u03b8<\/mi>\\theta<\/annotation><\/semantics><\/math>\u306e\u5834\u5408\u3001\u771f\u65b9\u4f4d\u306f\u901a\u5e38 90<\/mn>\u2218<\/mo><\/msup>\u2212<\/mo>\u03b8<\/mi><\/mrow>90^\\circ – \\theta<\/annotation><\/semantics><\/math>\u3067\u3059\u3002<\/p>\n\n\n\n

                                      [\u76f4\u4ea4\u5ea7\u6a19\uff08\u6771\u304b\u3089\u53cd\u6642\u8a08\u56de\u308a\uff09\u3068\u771f\u65b9\u4f4d\uff08\u5317\u304b\u3089\u6642\u8a08\u56de\u308a\uff09\u306e\u6bd4\u8f03\u753b\u50cf]<\/p>\n\n\n\n

                                      \n\n\n\n

                                      2. \u30b3\u30f3\u30d1\u30b9\uff08\u56db\u5206\u5100\uff09\u65b9\u4f4d<\/h3>\n\n\n\n

                                      \u3053\u308c\u306f\u3001\u5317\u307e\u305f\u306f\u5357\u304b\u3089\u6771\u307e\u305f\u306f\u897f\u306b\u5411\u304b\u3063\u3066\u300c\u56de\u8ee2\u300d\u3059\u308b\u3053\u3068\u3067\u65b9\u5411\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

                                        \n
                                      • \u5f62\u5f0f:<\/strong> [\u958b\u59cb\u65b9\u5411] [\u89d2\u5ea6] [\u56de\u8ee2\u65b9\u5411]<\/li>\n\n\n\n
                                      • \u4f8b:<\/strong> (\u5317\u304b\u3089\u958b\u59cb\u3001\u6771\u306b30<\/mn>\u2218<\/mo><\/msup>30^\\circ<\/annotation><\/semantics><\/math>\u56de\u8ee2) \u307e\u305f\u306f (\u5357\u304b\u3089\u958b\u59cb\u3001\u897f\u306b20<\/mn>\u2218<\/mo><\/msup>20^\\circ<\/annotation><\/semantics><\/math>\u56de\u8ee2)<\/li>\n<\/ul>\n\n\n\n
                                        \n\n\n\n

                                        \u5b9f\u4f8b: \u30d9\u30af\u30c8\u30eb\u306e\u5909\u63db<\/h3>\n\n\n\n

                                        \u5148\u307b\u3069\u4f7f\u7528\u3057\u305f\u30d9\u30af\u30c8\u30eb \ud835\udc2f<\/mi>=<\/mo>4<\/mn>\ud835\udc22<\/mi>+<\/mo>3<\/mn>\ud835\udc23<\/mi><\/mrow>\\mathbf{v} = 4\\mathbf{i} + 3\\mathbf{j}<\/annotation><\/semantics><\/math> \u3092\u4f7f\u7528\u3057\u307e\u3059: \u3002
                                        \u6570\u5b66\u7684\u306a\u89d2\u5ea6\u306f 36.87<\/mn>\u2218<\/mo><\/msup>36.87^\\circ<\/annotation><\/semantics><\/math> (x<\/mi>x<\/annotation><\/semantics><\/math>-\u8ef8\/\u6771\u304b\u3089\u6e2c\u5b9a) \u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

                                        \u771f\u65b9\u4f4d\u3092\u6c42\u3081\u308b\u306b\u306f:<\/h4>\n\n\n\n
                                          \n
                                        1. \u5317\u306f\u30b0\u30ea\u30c3\u30c9\u4e0a90<\/mn>\u2218<\/mo><\/msup>90^\\circ<\/annotation><\/semantics><\/math>\u306b\u3042\u308a\u307e\u3059\u3002<\/li>\n\n\n\n
                                        2. \u6771\u306f\u30b0\u30ea\u30c3\u30c9\u4e0a0<\/mn>\u2218<\/mo><\/msup>0^\\circ<\/annotation><\/semantics><\/math>\u306b\u3042\u308a\u307e\u3059\u304b\u3002<\/li>\n\n\n\n
                                        3. \u30d9\u30af\u30c8\u30eb\u306f\u6771\u304b\u3089 36.87<\/mn>\u2218<\/mo><\/msup>36.87^\\circ<\/annotation><\/semantics><\/math>\u300c\u4e0a\u300d\u306b\u3042\u308a\u307e\u3059\u3002<\/li>\n\n\n\n
                                        4. \u5317\u3068\u30d9\u30af\u30c8\u30eb\u306e\u5dee\u306f , 90<\/mn>\u2218<\/mo><\/msup>\u2212<\/mo>36.87<\/mn>\u2218<\/mo><\/msup>=<\/mo>53.13<\/mn>\u2218<\/mo><\/msup><\/mrow>90^\\circ – 36.87^\\circ = 53.13^\\circ<\/annotation><\/semantics><\/math>\u3002<\/li>\n\n\n\n
                                        5. \u771f\u65b9\u4f4d:<\/strong> 053.13<\/mn>\u2218<\/mo><\/msup>T<\/mi><\/mrow>053.13^\\circ T<\/annotation><\/semantics><\/math><\/li>\n<\/ol>\n\n\n\n

                                          \u30b3\u30f3\u30d1\u30b9\u65b9\u4f4d\u3092\u6c42\u3081\u308b\u306b\u306f:<\/h4>\n\n\n\n
                                            \n
                                          1. \u56f3\u3092\u898b\u3066\u304f\u3060\u3055\u3044\u3002\u30d9\u30af\u30c8\u30eb\u306f\u5317\u306b\u8fd1\u3044\u3067\u3059\u304b\u3001\u5357\u306b\u8fd1\u3044\u3067\u3059\u304b? \u5317<\/strong>\u3002<\/li>\n\n\n\n
                                          2. \u6771\u306b\u50be\u3044\u3066\u3044\u307e\u3059\u304b\u3001\u897f\u306b\u50be\u3044\u3066\u3044\u307e\u3059\u304b? \u6771<\/strong>\u3002<\/li>\n\n\n\n
                                          3. \u5317\u304b\u3089\u6771\u3078\u306e\u89d2\u5ea6\u306f 53.13<\/mn>\u2218<\/mo><\/msup>53.13^\\circ<\/annotation><\/semantics><\/math>\u3067\u3059\u3002<\/li>\n\n\n\n
                                          4. \u30b3\u30f3\u30d1\u30b9\u65b9\u4f4d:<\/strong> N<\/mi>53.13<\/mn>\u2218<\/mo><\/msup>E<\/mi><\/mrow>N 53.13^\\circ E<\/annotation><\/semantics><\/math><\/li>\n<\/ol>\n\n\n\n

                                            [\u7b2c1\u8c61\u9650\u306e\u30d9\u30af\u30c8\u30eb\u304c\u771f\u65b9\u4f4d\u3068\u30b3\u30f3\u30d1\u30b9\u65b9\u4f4d\u306e\u4e21\u65b9\u306b\u5909\u63db\u3055\u308c\u308b\u69d8\u5b50\u3092\u793a\u3059\u56f3]<\/p>\n\n\n\n

                                            \n\n\n\n

                                            4\u8c61\u9650\u306e\u6982\u8981\u8868<\/h3>\n\n\n\n
                                            \u8c61\u9650<\/th>\u30d9\u30af\u30c8\u30eb\u306e\u4f8b<\/th>\u76f4\u4ea4\u89d2 (\u03b8<\/mi>\\theta<\/annotation><\/semantics><\/math>)<\/th>\u771f\u65b9\u4f4d (T<\/mi>T<\/annotation><\/semantics><\/math>)<\/th><\/tr><\/thead>
                                            \u7b2c1\u8c61\u9650 (\u5317\u6771)<\/strong><\/td>4<\/mn>i<\/mi>+<\/mo>3<\/mn>j<\/mi><\/mrow>4i + 3j<\/annotation><\/semantics><\/math><\/td>36.9<\/mn>\u2218<\/mo><\/msup>36.9^\\circ<\/annotation><\/semantics><\/math><\/td>90<\/mn>\u2212<\/mo>36.9<\/mn>=<\/mo><\/mrow>90 – 36.9 = <\/annotation><\/semantics><\/math>
                                            \ud835\udfce\ud835\udfd3\ud835\udfd1.\ud835\udfcf<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow>\\mathbf{053.1^\\circ T}<\/annotation><\/semantics><\/math><\/td><\/tr>
                                            \u7b2c2\u8c61\u9650 (\u5317\u897f)<\/strong><\/td>\u2212<\/mo>4<\/mn>i<\/mi>+<\/mo>3<\/mn>j<\/mi><\/mrow>-4i + 3j<\/annotation><\/semantics><\/math><\/td>143.1<\/mn>\u2218<\/mo><\/msup>143.1^\\circ<\/annotation><\/semantics><\/math><\/td>450<\/mn>\u2212<\/mo>143.1<\/mn>=<\/mo><\/mrow>450 – 143.1 = <\/annotation><\/semantics><\/math>
                                            \ud835\udfd1\ud835\udfce\ud835\udfd4.\ud835\udfd7<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow>\\mathbf{306.9^\\circ T}<\/annotation><\/semantics><\/math>*<\/td><\/tr>
                                            \u7b2c3\u8c61\u9650 (\u5357\u897f)<\/strong><\/td>\u2212<\/mo>4<\/mn>i<\/mi>\u2212<\/mo>3<\/mn>j<\/mi><\/mrow>-4i – 3j<\/annotation><\/semantics><\/math><\/td>216.9<\/mn>\u2218<\/mo><\/msup>216.9^\\circ<\/annotation><\/semantics><\/math><\/td>450<\/mn>\u2212<\/mo>216.9<\/mn>=<\/mo><\/mrow>450 – 216.9 = <\/annotation><\/semantics><\/math>
                                            \ud835\udfd0\ud835\udfd1\ud835\udfd1.\ud835\udfcf<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow>\\mathbf{233.1^\\circ T}<\/annotation><\/semantics><\/math><\/td><\/tr>
                                            \u7b2c4\u8c61\u9650 (\u5357\u6771)<\/strong><\/td>4<\/mn>i<\/mi>\u2212<\/mo>3<\/mn>j<\/mi><\/mrow>4i – 3j<\/annotation><\/semantics><\/math><\/td>323.1<\/mn>\u2218<\/mo><\/msup>323.1^\\circ<\/annotation><\/semantics><\/math><\/td>450<\/mn>\u2212<\/mo>323.1<\/mn>=<\/mo><\/mrow>450 – 323.1 =<\/annotation><\/semantics><\/math>
                                            \ud835\udfcf\ud835\udfd0\ud835\udfd4.\ud835\udfd7<\/mn>\u2218<\/mo><\/msup>\ud835\udc13<\/mi><\/mrow> \\mathbf{126.9^\\circ T}<\/annotation><\/semantics><\/math><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

                                            *\u6ce8\uff1a\u771f\u65b9\u4f4d\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u7c21\u5358\u306a\u88cf\u6280\u306f <\/em>T<\/mi>r<\/mi>u<\/mi>e<\/mi>=<\/mo>(<\/mo>450<\/mn>\u2212<\/mo>\u03b8<\/mi>)<\/mo><\/mo><\/mspace>(<\/mo><\/mtext>mod<\/mi><\/mrow><\/mspace>360<\/mn>)<\/mo><\/mrow>True = (450 – \\theta) \\pmod{360}<\/annotation><\/semantics><\/math>\u3067\u3059\u3002\u3067\u3082\u6b63\u76f4\u306b\u8a00\u3046\u3068\uff1f \u7d75\u3092\u63cf\u3044\u3066\u300c\u5317\u300d\u306e\u7dda\u3092\u898b\u308b\u65b9\u304c\u305a\u3063\u3068\u5b89\u5168\u3067\u3059\uff01<\/em><\/p>\n\n\n\n
                                            \n\n\n\n

                                            \u8a66\u9a13\u5bfe\u7b56\u306e\u30d2\u30f3\u30c8<\/h3>\n\n\n\n

                                            \u554f\u984c\u306b \u300c\u65b9\u4f4d\u2026\u300d<\/strong> \u3068\u66f8\u304b\u308c\u3066\u3044\u305f\u3089\u3001\u3059\u3050\u306b\u300c\u5317\u304b\u3089\u6642\u8a08\u56de\u308a\u300d\u30e2\u30fc\u30c9\u306b\u5207\u308a\u66ff\u3048\u307e\u3057\u3087\u3046\u3002\u591a\u304f\u306e\u53d7\u9a13\u751f\u306f\u3001\u6b63\u3057\u3044\u89d2\u5ea6\u3092\u8a08\u7b97\u3057\u3066\u3082\u9593\u9055\u3063\u305f\u51fa\u767a\u70b9\u304b\u3089\u8a08\u7b97\u3057\u3066\u3057\u307e\u3046\u305f\u3081\u3001\u5931\u70b9\u3057\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

                                            \u300c\u7b2c\u4e09\u8c61\u9650\u300d\u30d9\u30af\u30c8\u30eb\uff08<\/strong>x<\/mi>x<\/annotation><\/semantics><\/math>\u3068<\/strong>y<\/mi>y<\/annotation><\/semantics><\/math>\u304c\u4e21\u65b9\u3068\u3082\u8ca0\uff09\u3092\u5909\u63db\u3057\u3066\u3001\u65b9\u4f4d\u3092\u6b63\u78ba\u306b\u8a08\u7b97\u3067\u304d\u308b\u304b\u8a66\u3057\u3066\u307f\u307e\u305b\u3093\u304b\uff1f<\/strong><\/p>\n\n\n\n

                                            ***************************************************************************<\/p>\n\n\n\n

                                            <\/p>\n","protected":false},"excerpt":{"rendered":"

                                            Welcome to Year 11! If you\u2019re diving into Specialist Mathematics or Mathematical Methods in Queensland (QCAA curriculum), vectors are about to become your new best friends. Think of a vector as a mathematical instruction. Until now, you\u2019ve mostly dealt with “scalars”\u2014numbers that only tell you how much (like 5 kg of apples or a temperature […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1201","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1201","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1201"}],"version-history":[{"count":8,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1201\/revisions"}],"predecessor-version":[{"id":1467,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1201\/revisions\/1467"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1201"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1201"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1201"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}