{"id":1393,"date":"2026-01-27T20:16:11","date_gmt":"2026-01-27T10:16:11","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1393"},"modified":"2026-01-28T19:52:39","modified_gmt":"2026-01-28T09:52:39","slug":"year12-math-4-3-3-vectors-in-three-dimensions","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1393","title":{"rendered":"Year12 MATH 4-3-3 vectors in three dimensions"},"content":{"rendered":"\n

In QLD Year 12 Specialist Mathematics Unit 3, the “Vectors in three dimensions” topic extends the two-dimensional vector concepts learned in Units 1 & 2 into three-dimensional (x<\/mi>,<\/mo>y<\/mi>,<\/mo>z<\/mi><\/mrow>x comma y comma z<\/annotation><\/semantics><\/math>) space. It focuses on representing, calculating, and applying vectors to solve complex geometric and physical problems, including the use of scalar (dot) products, vector (cross) products, and equations of lines and planes. <\/p>\n\n\n\n

Key concepts in this unit include: <\/p>\n\n\n\n

1. Representation and Algebra of 3D Vectors <\/strong><\/p>\n\n\n\n

    \n
  • 3D Cartesian Components:<\/strong> Vectors are represented in component form as v<\/mi>=<\/mo>a<\/mi>i<\/mi>+<\/mo>b<\/mi>j<\/mi>+<\/mo>c<\/mi>k<\/mi><\/mrow>bold v equals a bold i plus b bold j plus c bold k<\/annotation><\/semantics><\/math> or column vector form (<\/mo>a<\/mi><\/mtd><\/mtr>b<\/mi><\/mtd><\/mtr>c<\/mi><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>the 3 by 1 column matrix; a, b, c end-matrix;<\/annotation><\/semantics><\/math>, utilizing the three standard unit vectors i<\/mi>,<\/mo>j<\/mi>,<\/mo>k<\/mi><\/mrow>bold i comma bold j comma bold k<\/annotation><\/semantics><\/math> along the x<\/mi>,<\/mo>y<\/mi>,<\/mo><\/mrow>x comma y comma<\/annotation><\/semantics><\/math> and z<\/mi>z<\/annotation><\/semantics><\/math> axes.<\/li>\n\n\n\n
  • Magnitude and Direction:<\/strong> Calculating the magnitude of a 3D vector using the formula a<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>2<\/mn><\/msup>+<\/mo>c<\/mi>2<\/mn><\/msup><\/mrow><\/msqrt>the square root of a squared plus b squared plus c squared end-root<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  • Operations:<\/strong> Addition, subtraction, and scalar multiplication of 3D vectors. <\/li>\n<\/ul>\n\n\n\n

    2. Vector Products <\/strong><\/p>\n\n\n\n

      \n
    • Scalar (Dot) Product:<\/strong> Used to calculate the angle between two 3D vectors and determine if they are perpendicular (a<\/mi>\u22c5<\/mo>b<\/mi>=<\/mo>0<\/mn><\/mrow>bold a center dot bold b equals 0<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
    • Vector (Cross) Product:<\/strong> A crucial 3D-specific tool that produces a vector perpendicular to two given vectors (a<\/mi>\u00d7<\/mo>b<\/mi><\/mrow>bold a cross bold b<\/annotation><\/semantics><\/math>). It is used to find the area of parallelograms and triangles.<\/li>\n\n\n\n
    • Scalar Triple Product:<\/strong> Used for calculating the volume of parallelepipeds and identifying coplanar vectors. <\/li>\n<\/ul>\n\n\n\n

      3. Geometric Applications <\/strong><\/p>\n\n\n\n

        \n
      • Equations of Lines:<\/strong> Developing vector equations (r<\/mi>=<\/mo>a<\/mi>+<\/mo>\u03bb<\/mi>b<\/mi><\/mrow>bold r equals bold a plus lambda bold b<\/annotation><\/semantics><\/math>), parametric equations (x<\/mi>=<\/mo>x<\/mi>0<\/mn><\/msub>+<\/mo>\u03bb<\/mi>l<\/mi>,<\/mo>y<\/mi>=<\/mo>y<\/mi>0<\/mn><\/msub>+<\/mo>\u03bb<\/mi>m<\/mi>,<\/mo>z<\/mi>=<\/mo>z<\/mi>0<\/mn><\/msub>+<\/mo>\u03bb<\/mi>n<\/mi><\/mrow>x equals x sub 0 plus lambda l comma y equals y sub 0 plus lambda m comma z equals z sub 0 plus lambda n<\/annotation><\/semantics><\/math>), and Cartesian equations of lines in 3D space.<\/li>\n\n\n\n
      • Equations of Planes:<\/strong> Establishing equations of planes in the form r<\/mi>\u22c5<\/mo>n<\/mi>=<\/mo>d<\/mi><\/mrow>bold r center dot bold n equals d<\/annotation><\/semantics><\/math> or a<\/mi>x<\/mi>+<\/mo>b<\/mi>y<\/mi>+<\/mo>c<\/mi>z<\/mi>=<\/mo>d<\/mi><\/mrow>a x plus b y plus c z equals d<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
      • Intersection and Distance:<\/strong> Solving problems involving the intersection of lines and planes, and finding the distance from a point to a plane or line. <\/li>\n<\/ul>\n\n\n\n

        4. Motion and Physical Applications <\/strong><\/p>\n\n\n\n

          \n
        • Vectors in Motion:<\/strong> Describing position, velocity, and acceleration vectors as functions of time.<\/li>\n\n\n\n
        • Collision Detection:<\/strong> Determining if two objects moving in 3D space will collide.<\/li>\n\n\n\n
        • Applications:<\/strong> Modeling physical scenarios such as straight-line motion, projectile motion, and circular motion using vectors. <\/li>\n<\/ul>\n\n\n\n

          This topic often utilizes technology (CAS calculators) for complex computations. <\/p>\n\n\n\n

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

          QLD 12\u5e74\u751f\u5c02\u9580\u6570\u5b66\u30e6\u30cb\u30c3\u30c83\u306e\u300c3\u6b21\u5143\u306e\u30d9\u30af\u30c8\u30eb\u300d\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u30e6\u30cb\u30c3\u30c81\u30682\u3067\u5b66\u7fd2\u3057\u305f2\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u6982\u5ff5\u30923\u6b21\u5143((x,y,z))\u7a7a\u9593\u306b\u62e1\u5f35\u3057\u307e\u3059\u3002\u3053\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u30b9\u30ab\u30e9\u30fc\u7a4d\uff08\u5185\u7a4d\uff09\u3001\u30d9\u30af\u30c8\u30eb\u7a4d\uff08\u5916\u7a4d\uff09\u3001\u76f4\u7dda\u3068\u5e73\u9762\u306e\u65b9\u7a0b\u5f0f\u306e\u4f7f\u7528\u306a\u3069\u3001\u30d9\u30af\u30c8\u30eb\u306e\u8868\u73fe\u3001\u8a08\u7b97\u3001\u9069\u7528\u3092\u901a\u3057\u3066\u8907\u96d1\u306a\u5e7e\u4f55\u5b66\u304a\u3088\u3073\u7269\u7406\u5b66\u306e\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u306b\u7126\u70b9\u3092\u5f53\u3066\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

          \u3053\u306e\u5358\u5143\u306e\u4e3b\u8981\u306a\u6982\u5ff5\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n

            \n
          1. 3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u8868\u73fe\u3068\u4ee3\u6570<\/li>\n<\/ol>\n\n\n\n

            3\u6b21\u5143\u76f4\u4ea4\u5ea7\u6a19\u6210\u5206\uff1a\u30d9\u30af\u30c8\u30eb\u306f\u3001(x,y,)\u8ef8\u3068(z)\u8ef8\u306b\u6cbf\u3063\u305f3\u3064\u306e\u6a19\u6e96\u5358\u4f4d\u30d9\u30af\u30c8\u30eb(\ud835\udc22<\/mi>,<\/mo>\ud835\udc23<\/mi>,<\/mo>\ud835\udc24<\/mi><\/mrow>\\mathbf{i},\\mathbf{j},\\mathbf{k}<\/annotation><\/semantics><\/math>)\u3092\u7528\u3044\u3066\u3001\u6210\u5206\u5f62\u5f0f(\ud835\udc2f<\/mi>=<\/mo>a<\/mi>\ud835\udc22<\/mi>+<\/mo>b<\/mi>\ud835\udc23<\/mi>+<\/mo>c<\/mi>\ud835\udc24<\/mi><\/mrow>\\mathbf{v}=a\\mathbf{i}+b\\mathbf{j}+c\\mathbf{k}<\/annotation><\/semantics><\/math>)\u307e\u305f\u306f\u5217\u30d9\u30af\u30c8\u30eb\u5f62\u5f0f (<\/mo>a<\/mi><\/mtd><\/mtr>b<\/mi><\/mtd><\/mtr>c<\/mi><\/mtd><\/mtr><\/mtable>)<\/mo><\/mrow>\\left(\\begin{matrix}a\\\\ b\\\\ c\\end{matrix}\\right)<\/annotation><\/semantics><\/math>\u3067\u8868\u308f\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

            \u5927\u304d\u3055\u3068\u65b9\u5411\uff1a3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u306f\u3001\u5f0f(a<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>2<\/mn><\/msup>+<\/mo>c<\/mi>2<\/mn><\/msup><\/mrow><\/msqrt>\\sqrt{a^{2}+b^{2}+c^{2}}<\/annotation><\/semantics><\/math>)\u3092\u7528\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

            \u6f14\u7b97\uff1a3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u52a0\u7b97\u3001\u6e1b\u7b97\u3001\u30b9\u30ab\u30e9\u30fc\u4e57\u7b97\u3002<\/p>\n\n\n\n

              \n
            1. \u30d9\u30af\u30c8\u30eb\u7a4d<\/li>\n<\/ol>\n\n\n\n

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              \u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\uff1a\u5e73\u884c\u516d\u9762\u4f53\u306e\u4f53\u7a4d\u3092\u8a08\u7b97\u3057\u3001\u5171\u9762\u30d9\u30af\u30c8\u30eb\u3092\u8b58\u5225\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

                \n
              1. \u5e7e\u4f55\u5b66\u306e\u5fdc\u7528<\/li>\n<\/ol>\n\n\n\n

                \u76f4\u7dda\u306e\u65b9\u7a0b\u5f0f\uff1a\u30d9\u30af\u30c8\u30eb\u65b9\u7a0b\u5f0f (\ud835\udc2b<\/mi>=<\/mo>\ud835\udc1a<\/mi>+<\/mo>\u03bb<\/mi>\ud835\udc1b<\/mi><\/mrow>\\mathbf{r}=\\mathbf{a}+\\lambda \\mathbf{b}<\/annotation><\/semantics><\/math>)\u3001\u5a92\u4ecb\u5909\u6570\u65b9\u7a0b\u5f0f (x<\/mi>=<\/mo>x<\/mi>0<\/mn><\/msub>+<\/mo>\u03bb<\/mi>l<\/mi>,<\/mo>y<\/mi>=<\/mo>y<\/mi>0<\/mn><\/msub>+<\/mo>\u03bb<\/mi>m<\/mi>,<\/mo>z<\/mi>=<\/mo>z<\/mi>0<\/mn><\/msub>+<\/mo>\u03bb<\/mi>n<\/mi><\/mrow>x=x_{0}+\\lambda l,y=y_{0}+\\lambda m,z=z_{0}+\\lambda n<\/annotation><\/semantics><\/math>)\u3001\u304a\u3088\u30733\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u76f4\u7dda\u306e\u76f4\u4ea4\u5ea7\u6a19\u65b9\u7a0b\u5f0f\u3092\u69cb\u7bc9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

                \u5e73\u9762\u306e\u65b9\u7a0b\u5f0f\uff1a(\ud835\udc2b<\/mi>\u22c5<\/mo>\ud835\udc27<\/mi>=<\/mo>d<\/mi><\/mrow>\\mathbf{r}\\cdot \\mathbf{n}=d<\/annotation><\/semantics><\/math>) \u307e\u305f\u306f (a<\/mi>x<\/mi>+<\/mo>b<\/mi>y<\/mi>+<\/mo>c<\/mi>z<\/mi>=<\/mo>d<\/mi><\/mrow>ax+by+cz=d<\/annotation><\/semantics><\/math>) \u306e\u5f62\u5f0f\u3067\u5e73\u9762\u306e\u65b9\u7a0b\u5f0f\u3092\u69cb\u7bc9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

                \u4ea4\u5dee\u3068\u8ddd\u96e2\uff1a\u76f4\u7dda\u3068\u5e73\u9762\u306e\u4ea4\u5dee\u306b\u95a2\u3059\u308b\u554f\u984c\u3092\u89e3\u304d\u3001\u70b9\u304b\u3089\u5e73\u9762\u307e\u305f\u306f\u76f4\u7dda\u307e\u3067\u306e\u8ddd\u96e2\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n

                  \n
                1. \u904b\u52d5\u3068\u7269\u7406\u7684\u5fdc\u7528<\/li>\n<\/ol>\n\n\n\n

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                  \u885d\u7a81\u691c\u51fa\uff1a3\u6b21\u5143\u7a7a\u9593\u3092\u79fb\u52d5\u3059\u308b2\u3064\u306e\u7269\u4f53\u304c\u885d\u7a81\u3059\u308b\u304b\u3069\u3046\u304b\u3092\u5224\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n

                  \u5fdc\u7528\uff1a\u76f4\u7dda\u904b\u52d5\u3001\u6295\u5c04\u904b\u52d5\u3001\u5186\u904b\u52d5\u306a\u3069\u306e\u7269\u7406\u7684\u30b7\u30ca\u30ea\u30aa\u3092\u30d9\u30af\u30c8\u30eb\u3092\u7528\u3044\u3066\u30e2\u30c7\u30eb\u5316\u3059\u308b\u3002<\/p>\n\n\n\n

                  \u3053\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u8907\u96d1\u306a\u8a08\u7b97\u306e\u305f\u3081\u306bCAS\u8a08\u7b97\u6a5f\u306a\u3069\u306e\u30c6\u30af\u30ce\u30ed\u30b8\u30fc\u304c\u3088\u304f\u5229\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

                  In QLD Year 12 Specialist Mathematics Unit 3, the “Vectors in three dimensions” topic extends the two-dimensional vector concepts learned in Units 1 & 2 into three-dimensional (x,y,zx comma y comma z) space. It focuses on representing, calculating, and applying vectors to solve complex geometric and physical problems, including the use of scalar (dot) products, […]<\/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-1393","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1393","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=1393"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1393\/revisions"}],"predecessor-version":[{"id":1458,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1393\/revisions\/1458"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1393"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1393"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}