{"id":1468,"date":"2026-01-31T13:04:15","date_gmt":"2026-01-31T03:04:15","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1468"},"modified":"2026-01-31T13:04:15","modified_gmt":"2026-01-31T03:04:15","slug":"year11-math-4-1-21dot-product","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1468","title":{"rendered":"Year11 MATH 4-1-21Dot Product"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The dot product (or scalar product) of two vectors,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mo>\u27e8<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo>\u27e9<\/mo><\/mrow><annotation encoding=\"text\/plain\">modified a with right arrow above equals open angle bracket a sub 1 comma a sub 2 close angle bracket<\/annotation><\/semantics><\/math> and  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mo>\u27e8<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo>\u27e9<\/mo><\/mrow><annotation encoding=\"text\/plain\">modified b with right arrow above equals open angle bracket b sub 1 comma b sub 2 close angle bracket<\/annotation><\/semantics><\/math>,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is a scalar value calculated as   <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>a<\/mi><mn>1<\/mn><\/msub><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><msub><mi>b<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"text\/plain\">a sub 1 b sub 1 plus a sub 2 b sub 2<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It measures the alignment of vectors, with   <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mo>|<\/mo><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>|<\/mo><mo>|<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>|<\/mo><mi>cos<\/mi><mo>(<\/mo><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">modified a with right arrow above center dot modified b with right arrow above equals the absolute value of modified a with right arrow above end-absolute-value the absolute value of modified b with right arrow above end-absolute-value cosine open paren theta close paren<\/annotation><\/semantics><\/math>, and is zero if they are perpendicular.\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Definitions and Formulas<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Algebraic Definition:<\/strong> The sum of the products of corresponding components:  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mo>\u2026<\/mo><mo>+<\/mo><msub><mi>a<\/mi><mi>n<\/mi><\/msub><msub><mi>b<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">modified a with right arrow above center dot modified b with right arrow above equals a sub 1 b sub 1 plus a sub 2 b sub 2 plus \u2026 plus a sub n b sub n<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Geometric Definition:<\/strong> The product of the magnitudes of the vectors and the cosine of the angle  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>\u03b8<\/mi><annotation encoding=\"text\/plain\">theta<\/annotation><\/semantics><\/math> between them:  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mo>|<\/mo><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>|<\/mo><mo>|<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>|<\/mo><mi>cos<\/mi><mo>(<\/mo><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">modified a with right arrow above center dot modified b with right arrow above equals the absolute value of modified a with right arrow above end-absolute-value the absolute value of modified b with right arrow above end-absolute-value cosine open paren theta close paren<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Vector Magnitude:<\/strong> A vector&#8217;s dot product with itself equals the square of its magnitude:  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>v<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>v<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mo>|<\/mo><mover accent=\"true\"><mi>v<\/mi><mo>\u20d7<\/mo><\/mover><msup><mo>|<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"text\/plain\">modified v with right arrow above center dot modified v with right arrow above equals the absolute value of modified v with right arrow above end-absolute-value squared<\/annotation><\/semantics><\/math>.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Properties<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Scalar Result:<\/strong> The output is a number, not a vector.<\/li>\n\n\n\n<li><strong>Commutative:<\/strong>  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><\/mrow><annotation encoding=\"text\/plain\">modified a with right arrow above center dot modified b with right arrow above equals modified b with right arrow above center dot modified a with right arrow above<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Distributive:<\/strong>  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mo>(<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>+<\/mo><mover accent=\"true\"><mi>c<\/mi><mo>\u20d7<\/mo><\/mover><mo>)<\/mo><mo>=<\/mo><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>+<\/mo><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>c<\/mi><mo>\u20d7<\/mo><\/mover><\/mrow><annotation encoding=\"text\/plain\">modified a with right arrow above center dot open paren modified b with right arrow above plus modified c with right arrow above close paren equals modified a with right arrow above center dot modified b with right arrow above plus modified a with right arrow above center dot modified c with right arrow above<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Orthogonal Vectors:<\/strong> If the dot product is 0, the vectors are perpendicular ( <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"text\/plain\">theta equals 90 raised to the composed with power<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Parallel Vectors:<\/strong> The dot product is maximized when vectors point in the same direction ( <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"text\/plain\">theta equals 0 raised to the composed with power<\/annotation><\/semantics><\/math>).\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Common Uses<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Finding Angles:<\/strong>  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>cos<\/mi><mo>(<\/mo><mi>\u03b8<\/mi><mo>)<\/mo><mo>=<\/mo><mfrac><mrow><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><\/mrow><mrow><mo>|<\/mo><mover accent=\"true\"><mi>a<\/mi><mo>\u20d7<\/mo><\/mover><mo>|<\/mo><mo>|<\/mo><mover accent=\"true\"><mi>b<\/mi><mo>\u20d7<\/mo><\/mover><mo>|<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">cosine open paren theta close paren equals the fraction with numerator modified a with right arrow above center dot modified b with right arrow above and denominator the absolute value of modified a with right arrow above end-absolute-value the absolute value of modified b with right arrow above end-absolute-value end-fraction<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Physics:<\/strong> Calculating work done by a force,  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>W<\/mi><mo>=<\/mo><mover accent=\"true\"><mi>F<\/mi><mo>\u20d7<\/mo><\/mover><mo>\u22c5<\/mo><mover accent=\"true\"><mi>d<\/mi><mo>\u20d7<\/mo><\/mover><\/mrow><annotation encoding=\"text\/plain\">cap W equals modified cap F with right arrow above center dot modified d with right arrow above<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Projections:<\/strong> Finding the projection of one vector onto another.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example Calculation<\/strong><br>If <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>u<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mo>\u27e8<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>\u27e9<\/mo><\/mrow><annotation encoding=\"text\/plain\">modified u with right arrow above equals open angle bracket 2 comma 3 close angle bracket<\/annotation><\/semantics><\/math> and  <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mi>v<\/mi><mo>\u20d7<\/mo><\/mover><mo>=<\/mo><mo>\u27e8<\/mo><mn>4<\/mn><mo>,<\/mo><mn>-1<\/mn><mo>\u27e9<\/mo><\/mrow><annotation encoding=\"text\/plain\">modified v with right arrow above equals open angle bracket 4 comma negative 1 close angle bracket<\/annotation><\/semantics><\/math>, then: <math data-latex=\"\\vec{u}\\cdot \\vec{v}=(2\\times 4)+(3\\times -1)=8-3=5\"><semantics><mrow><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>v<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>8<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{u}\\cdot \\vec{v}=(2\\times 4)+(3\\times -1)=8-3=5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">*****************************************************************************************<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u30c9\u30c3\u30c8\u7a4d\uff08\u307e\u305f\u306f\u30b9\u30ab\u30e9\u30fc\u7a4d\uff09<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">2\u3064\u306e\u30d9\u30af\u30c8\u30eb <math data-latex=\"\\vec{a}=\\langle a_{1},a_{2}\\rangle \"><semantics><mrow><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u27e8<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">\u27e9<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{a}=\\langle a_{1},a_{2}\\rangle <\/annotation><\/semantics><\/math>\u3068\u3001<math data-latex=\"\\vec{b}=\\langle b_{1},b_{2}\\rangle \"><semantics><mrow><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u27e8<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">\u27e9<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{b}=\\langle b_{1},b_{2}\\rangle <\/annotation><\/semantics><\/math>  \u306e\u30c9\u30c3\u30c8\u7a4d\uff08\u307e\u305f\u306f\u30b9\u30ab\u30e9\u30fc\u7a4d\uff09\u306f\u3001<math data-latex=\"a_{1}b_{1}+a_{2}b_{2}\"><semantics><mrow><msub><mi>a<\/mi><mn>1<\/mn><\/msub><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><msub><mi>b<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">a_{1}b_{1}+a_{2}b_{2}<\/annotation><\/semantics><\/math> \u3068\u3057\u3066\u8a08\u7b97\u3055\u308c\u308b\u30b9\u30ab\u30e9\u30fc\u5024\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u308c\u306f\u30d9\u30af\u30c8\u30eb\u306e\u914d\u7f6e\u3092 <math data-latex=\"\\vec{a}\\cdot \\vec{b}=|\\vec{a}||\\vec{b}|\\cos (\\theta )\"><semantics><mrow><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mi>|<\/mi><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mi>|<\/mi><mi>|<\/mi><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mi>|<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{a}\\cdot \\vec{b}=|\\vec{a}||\\vec{b}|\\cos (\\theta )<\/annotation><\/semantics><\/math> \u3067\u6e2c\u5b9a\u3057\u3001\u30d9\u30af\u30c8\u30eb\u304c\u76f4\u4ea4\u3059\u308b\u5834\u5408\u306f 0 \u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u8981\u306a\u5b9a\u7fa9\u3068\u516c\u5f0f<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ee3\u6570\u7684\u5b9a\u7fa9\uff1a\u5bfe\u5fdc\u3059\u308b\u6210\u5206\u306e\u7a4d\u306e\u548c\uff1a<math data-latex=\"a\u20d7\u22c5b\u20d7=a1b1+a2b2+\u2026+anbn\"><semantics><mrow><mi>a<\/mi><mtext>\u20d7<\/mtext><mo>\u22c5<\/mo><mi>b<\/mi><mtext>\u20d7<\/mtext><mo>=<\/mo><mi>a<\/mi><mn>1<\/mn><mi>b<\/mi><mn>1<\/mn><mo>+<\/mo><mi>a<\/mi><mn>2<\/mn><mi>b<\/mi><mn>2<\/mn><mo>+<\/mo><mo>\u2026<\/mo><mo>+<\/mo><mi>a<\/mi><mi>n<\/mi><mi>b<\/mi><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a\u20d7\u22c5b\u20d7=a1b1+a2b2+\u2026+anbn<\/annotation><\/semantics><\/math> <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5e7e\u4f55\u5b66\u7684\u5b9a\u7fa9\uff1a\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3068\u305d\u308c\u3089\u306e\u9593\u306e\u89d2\u5ea6 \u03b8 \u306e\u4f59\u5f26\u306e\u7a4d\uff1a<math data-latex=\"\\vec{a}\\cdot \\vec{b}=|\\vec{a}||\\vec{b}|\\cos (\\theta )\"><semantics><mrow><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mi>|<\/mi><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mi>|<\/mi><mi>|<\/mi><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mi>|<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{a}\\cdot \\vec{b}=|\\vec{a}||\\vec{b}|\\cos (\\theta )<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\uff1a\u30d9\u30af\u30c8\u30eb\u3068\u305d\u308c\u81ea\u8eab\u306e\u5185\u7a4d\u306f\u3001\u305d\u306e\u5927\u304d\u3055\u306e2\u4e57\u306b\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002<math data-latex=\" \\vec{v}\\cdot \\vec{v}=|\\vec{v}|^{2}\"><semantics><mrow><mover><mi>v<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>v<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mi>|<\/mi><mover><mi>v<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><msup><mi>|<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\"> \\vec{v}\\cdot \\vec{v}=|\\vec{v}|^{2}<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u306a\u7279\u6027<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b9\u30ab\u30e9\u30fc\u7d50\u679c\uff1a\u51fa\u529b\u306f\u30d9\u30af\u30c8\u30eb\u3067\u306f\u306a\u304f\u6570\u5024\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ea4\u63db\u6cd5\u5247\uff1a<math data-latex=\"\\vec{a}\\cdot \\vec{b}=\\vec{b}\\cdot \\vec{a}\"><semantics><mrow><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{a}\\cdot \\vec{b}=\\vec{b}\\cdot \\vec{a}<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5206\u914d\u6cd5\u5247: <math data-latex=\"\\vec{a}\\cdot (\\vec{b}+\\vec{c})=\\vec{a}\\cdot \\vec{b}+\\vec{a}\\cdot \\vec{c}\"><semantics><mrow><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>+<\/mo><mover><mi>c<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>+<\/mo><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>c<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{a}\\cdot (\\vec{b}+\\vec{c})=\\vec{a}\\cdot \\vec{b}+\\vec{a}\\cdot \\vec{c}<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u76f4\u4ea4\u30d9\u30af\u30c8\u30eb: \u5185\u7a4d (dot product) \u304c 0 \u306e\u5834\u5408\u3001\u30d9\u30af\u30c8\u30eb\u306f\u76f4\u4ea4\u3057\u307e\u3059 (<math data-latex=\" \\theta =90^{\\circ }\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><msup><mn>90<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\"> \\theta =90^{\\circ }<\/annotation><\/semantics><\/math>)\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5e73\u884c\u30d9\u30af\u30c8\u30eb: \u5185\u7a4d\u306f\u3001\u30d9\u30af\u30c8\u30eb\u304c\u540c\u3058\u65b9\u5411\u3092\u5411\u3044\u3066\u3044\u308b\u3068\u304d\u306b\u6700\u5927\u306b\u306a\u308a\u307e\u3059 (<math data-latex=\"\\theta =90^{\\circ }\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><msup><mn>90<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\theta =90^{\\circ }<\/annotation><\/semantics><\/math>)\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e00\u822c\u7684\u306a\u7528\u9014<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u89d2\u5ea6\u306e\u8a08\u7b97: <math data-latex=\"\\cos (\\theta )=\\frac{\\vec{a}\\cdot \\vec{b}}{|\\vec{a}||\\vec{b}|}\"><semantics><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><mrow><mi>|<\/mi><mover><mi>a<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mi>|<\/mi><mi>|<\/mi><mover><mi>b<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mi>|<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos (\\theta )=\\frac{\\vec{a}\\cdot \\vec{b}}{|\\vec{a}||\\vec{b}|}<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7269\u7406\u5b66: \u529b\u306b\u3088\u3063\u3066\u306a\u3055\u308c\u305f\u4ed5\u4e8b\u306e\u8a08\u7b97       <math data-latex=\"W=\\vec{F}\\cdot \\vec{d}\"><semantics><mrow><mi>W<\/mi><mo>=<\/mo><mover><mi>F<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>d<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">W=\\vec{F}\\cdot \\vec{d}<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6295\u5f71: \u3042\u308b\u30d9\u30af\u30c8\u30eb\u304b\u3089\u5225\u306e\u30d9\u30af\u30c8\u30eb\u3078\u306e\u6295\u5f71\u3092\u6c42\u3081\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8a08\u7b97\u4f8b :  <math data-latex=\"\\vec{u}=\\langle 2,3\\rangle \"><semantics><mrow><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u27e8<\/mo><mn>2,3<\/mn><mo form=\"postfix\" stretchy=\"false\">\u27e9<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{u}=\\langle 2,3\\rangle <\/annotation><\/semantics><\/math> and <math data-latex=\"\\vec{v}=\\langle 4,-1\\rangle \"><semantics><mrow><mover><mi>v<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u27e8<\/mo><mn>4<\/mn><mo separator=\"true\">,<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">\u27e9<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{v}=\\langle 4,-1\\rangle <\/annotation><\/semantics><\/math> \u306e\u3068\u304d\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3000<math data-latex=\"\\vec{u}\\cdot \\vec{v}=(2\\times 4)+(3\\times -1)=8-3=5\"><semantics><mrow><mover><mi>u<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>\u22c5<\/mo><mover><mi>v<\/mi><mo stretchy=\"false\" style=\"transform:scale(0.75) translate(10%, 30%);\">\u2192<\/mo><\/mover><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>8<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{u}\\cdot \\vec{v}=(2\\times 4)+(3\\times -1)=8-3=5<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The dot product (or scalar product) of two vectors, a\u20d7=\u27e8a1,a2\u27e9modified a with right arrow above equals open angle bracket a sub 1 comma a sub 2 close angle bracket and b\u20d7=\u27e8b1,b2\u27e9modified b with right arrow above equals open angle bracket b sub 1 comma b sub 2 close angle bracket, is a scalar value calculated [&hellip;]<\/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-1468","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1468","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=1468"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1468\/revisions"}],"predecessor-version":[{"id":1471,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1468\/revisions\/1471"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1468"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1468"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1468"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}