\u5411\u91cf<\/h2>\n\n\n\n
\u5411\u91cf\u957f\u5ea6\uff1a[latex]\\left | \\overrightarrow{a} \\right |[\/latex]<\/p>\n\n\n\n
\u5355\u4f4d\u5411\u91cf\uff1a[latex]\\widehat{a}= \\frac{\\overrightarrow{a}}{\\left | \\overrightarrow{a} \\right | }[\/latex]<\/p>\n\n\n\n
[latex]A=\\begin{bmatrix}x\\\\y\\end{bmatrix}[\/latex]
<\/p>\n\n\n\n
[latex]A^{T} =\\begin{bmatrix}x&y\\end{bmatrix}[\/latex]
<\/p>\n\n\n\n
[latex]\\left | A \\right | =\\sqrt{x^2+y^2}[\/latex]
<\/p>\n\n\n\n
\u5411\u91cf\u7684\u4e58\u6cd5<\/h2>\n\n\n\n\u70b9\u4e58<\/h3>\n\n\n\n\u8fd0\u7b97<\/h4>\n\n\n\n
\u8fd0\u7b97<\/h4>\n\n\n\n
[latex]\\overrightarrow{a}\\cdot \\overrightarrow{b}=\\left | \\overrightarrow{a} \\right | \\left | \\overrightarrow{b} \\right | cos\\theta[\/latex]
<\/p>\n\n\n\n
[latex]cos\\theta =\\frac{ \\overrightarrow{a}\\cdot \\overrightarrow{b}}{\\left | \\overrightarrow{a} \\right | \\left | \\overrightarrow{b} \\right | }[\/latex]
<\/p>\n\n\n\n
\u5bf9\u4e8e\u5355\u4f4d\u5411\u91cf\uff1a[latex]cos\\theta =\\widehat{a}\\cdot\\widehat{b}[\/latex]
<\/p>\n\n\n\n
[latex] \\overrightarrow{a}\\cdot \\overrightarrow{b} = \\overrightarrow{b}\\cdot \\overrightarrow{a} [\/latex]
<\/p>\n\n\n\n
[latex]\\overrightarrow{a}\\cdot \\left ( \\overrightarrow{b}+\\overrightarrow{c}\\right ) =\\overrightarrow{a}\\cdot\\overrightarrow{b}+\\overrightarrow{a}\\cdot\\overrightarrow{c}[\/latex]
<\/p>\n\n\n\n
[latex]\\left ( k\\overrightarrow{a}\\right )\\cdot \\overrightarrow{b} =\\overrightarrow{a}\\cdot \\left ( k\\overrightarrow{b}\\right )=k\\left (\\overrightarrow{a}\\cdot \\overrightarrow{b}\\right )[\/latex]
<\/p>\n\n\n\n
In 2D\uff1a[latex]\\overrightarrow{a}\\cdot \\overrightarrow{b}=\\begin{bmatrix}x_{a} \\\\y_{a}\\end{bmatrix}\\cdot \\begin{bmatrix}x_{b} \\\\y_{b}\\end{bmatrix}=x_{a}x_{b}+y_{a}y_{b}[\/latex]<\/p>\n\n\n\n
In 3D\uff1a[latex]\\overrightarrow{a}\\cdot \\overrightarrow{b}=\\begin{bmatrix}x_{a} \\\\y_{a}\\\\z_{a}\\end{bmatrix}\\cdot \\begin{bmatrix}x_{b} \\\\y_{b}\\\\z_{b}\\end{bmatrix}=x_{a}x_{b}+y_{a}y_{b}+z_{a}z_{b}[\/latex]<\/p>\n\n\n\n
\u6295\u5f71<\/h4>\n\n\n\n
[latex]\\overrightarrow {b}_{\\bot}[\/latex]\uff1ab\u5230a\u7684\u6295\u5f71<\/p>\n\n\n\n
[latex]\\overrightarrow {b}_{\\bot}[\/latex] \u5fc5\u987b\u6cbf\u7740[latex]\\overrightarrow{a} [\/latex]\u6216 [latex]\\widehat{a} [\/latex] <\/p>\n\n\n\n
\u5373 [latex]\\overrightarrow {b}_{\\bot}=k\\widehat{a} [\/latex] <\/p>\n\n\n\n
[latex]k=\\left | \\overrightarrow {b}_{\\bot} \\right | = \\left | \\overrightarrow {b} \\right | cos\\theta [\/latex]
<\/p>\n\n\n\n
\u70b9\u4e58\u5728\u56fe\u5f62\u5b66\u4e2d\u7684\u5e94\u7528<\/h4>\n\n\n\n
\u6c42\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u4f8b\u5982\u5149\u6e90\u548c\u8868\u9762\u4e4b\u95f4\u89d2\u5ea6\u7684\u4f59\u5f26\u3002<\/p>\n\n\n\n
\u5bfb\u627e\u4e00\u4e2a\u5411\u91cf\u5728\u53e6\u4e00\u4e2a\u5411\u91cf\u4e0a\u7684\u6295\u5f71\u3002<\/p>\n\n\n\n
\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u6709\u591a\u8fd1\u3002<\/p>\n\n\n\n
\u5206\u89e3\u5411\u91cf\uff1a
\u6cbf [latex]\\overrightarrow{a} [\/latex] \u65b9\u5411 [latex]\\overrightarrow {b}_{\\bot}[\/latex]
\u548c\u5782\u76f4\u4e8e [latex]\\overrightarrow{a} [\/latex] \u65b9\u5411\uff1a [latex] \\overrightarrow {b} -\\overrightarrow {b}_{\\bot}[\/latex]<\/p>\n\n\n\n
\u786e\u5b9a\u5411\u91cf\u65b9\u5411\uff1a
\u70b9\u4e58\u7ed3\u679c>0\u540c\u5411
<0\u53cd\u5411
=0\u5782\u76f4<\/p>\n\n\n\n
\u53c9\u4e58<\/h3>\n\n\n\n\u53c9\u4e58\u7684\u8fd0\u7b97<\/h4>\n\n\n\n
\u53c9\u79ef\u4e0e\u4e24\u4e2a\u521d\u59cb\u5411\u91cf\u6b63\u4ea4<\/p>\n\n\n\n
\u53f3\u624b\u87ba\u65cb\u5b9a\u5219\u51b3\u5b9a\u65b9\u5411<\/p>\n\n\n\n
[latex]\\left |a \\times b\\right |=\\left|a\\right| \\left|b\\right| \\sin \\phi[\/latex]
<\/p>\n\n\n\n
[latex]\\vec{x} \\times \\vec{y}=+\\vec{z}[\/latex]
[latex]\\vec{y} \\times \\vec{x}=-\\vec{z}[\/latex]
[latex]\\vec{y} \\times \\vec{z}=+\\vec{x}[\/latex]
[latex]\\vec{z} \\times \\vec{y}=-\\vec{x}[\/latex]
[latex]\\vec{z} \\times \\vec{x}=+\\vec{y}[\/latex]
[latex]\\vec{x} \\times \\vec{z}=-\\vec{y}[\/latex]
<\/p>\n\n\n\n
[latex]\\vec{a} \\times \\vec{b}=-\\vec{b} \\times \\vec{a}[\/latex]
[latex]\\vec{a} \\times \\vec{a}=\\overrightarrow{0}[\/latex]
[latex]\\vec{a} \\times(\\vec{b}+\\vec{c})=\\vec{a} \\times \\vec{b}+\\vec{a} \\times \\vec{c}[\/latex]
[latex]\\vec{a} \\times(k \\vec{b})=k(\\vec{a} \\times \\vec{b})[\/latex]<\/p>\n\n\n\n
[latex]\\vec{a} \\times \\vec{b}=\\begin{bmatrix}
y_a z_b-y_b z_a \\\\
z_a x_b-x_a z_b \\\\
x_a y_b-y_a x_b
\\end{bmatrix}[\/latex]
<\/p>\n\n\n\n
[latex]\\vec{a} \\times \\vec{b}=A^* b=\\begin{bmatrix}
0 & -z_a & y_a \\\\
z_a & 0 & -x_a \\\\
-y_a & x_a & 0
\\end{bmatrix}\\begin{bmatrix}
x_b \\\\
y_b \\\\
z_b
\\end{bmatrix}[\/latex]
<\/p>\n\n\n\n
\u70b9\u4e58\u5728\u56fe\u5f62\u5b66\u4e2d\u7684\u5e94\u7528 <\/h4>\n\n\n\n\u786e\u5b9a\u5de6\/\u53f3<\/h5>\n\n\n\n
\u53c9\u79ef\u4e3a\u6b63\u5728\u5de6\u8fb9\uff0c\u53c9\u79ef\u4e3a\u8d1f\u5728\u53f3\u8fb9<\/p>\n\n\n\n![\"\"](\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2023\/10\/image.png\")
![\"\"](\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2023\/10\/image-1.png\")