向量的坐标运算

教材原文

$\begin{array}{r l} \boldsymbol {a} + \boldsymbol {b} & = (x _ {1} \boldsymbol {i} + y _ {1} \boldsymbol {j}) + (x _ {2} \boldsymbol {i} + y _ {2} \boldsymbol {j}) \\ & = x _ {1} \boldsymbol {i} + x _ {2} \boldsymbol {i} + y _ {1} \boldsymbol {j} + y _ {2} \boldsymbol {j} \\ & = (x _ {1} + x _ {2}) \boldsymbol {i} + (y _ {1} + y _ {2}) \boldsymbol {j}, \end{array}$

即

$\boldsymbol {a} + \boldsymbol {b} = (x _ {1} + x _ {2}, y _ {1} + y _ {2}).$

同理可得

$\boldsymbol {a} - \boldsymbol {b} = (x _ {1} - x _ {2}, y _ {1} - y _ {2}).$

这就是说，两个向量和（差）的坐标分别等于这两个向量相应坐标的和（差）.

如图6.3-12，作向量 $\overrightarrow{OA}$ ， $\overrightarrow{OB}$ ，则

$\begin{array}{r l} \overrightarrow {A B} & = \overrightarrow {O B} - \overrightarrow {O A} \\ & = (x _ {2}, y _ {2}) - (x _ {1}, y _ {1}) \\ & = (x _ {2} - x _ {1}, y _ {2} - y _ {1}). \end{array}$

因此，一个向量的坐标等于表示此向量的有向线段的终点的坐标减去起点的坐标.

$\lambda \boldsymbol {a} = \lambda (x \boldsymbol {i} + y \boldsymbol {j}) = \lambda x \boldsymbol {i} + \lambda y \boldsymbol {j},$

即

$\lambda \boldsymbol {a} = (\lambda x, \lambda y).$

这就是说，实数与向量的积的坐标等于用这个实数乘原来向量的相应坐标.

因为 $a = x_{1}i + y_{1}j, b = x_{2}i + y_{2}j,$

$\boldsymbol {a} \cdot \boldsymbol {b} = \left(x _ {1} \boldsymbol {i} + y _ {1} \boldsymbol {j}\right) \cdot \left(x _ {2} \boldsymbol {i} + y _ {2} \boldsymbol {j}\right) = x _ {1} x _ {2} \boldsymbol {i} ^ {2} + x _ {1} y _ {2} \boldsymbol {i} \cdot \boldsymbol {j} + y _ {1} x _ {2} \boldsymbol {j} \cdot \boldsymbol {i} + y _ {1} y _ {2} \boldsymbol {j} ^ {2}$

又 $i \cdot i = 1, j \cdot j = 1, i \cdot j = j \cdot i = 0,$

所以 $a \cdot b = x_{1}x_{2} + y_{1}y_{2}.$

这就是说，两个向量的数量积等于它们对应坐标的乘积的和.

由此可得

(1) 若 $\boldsymbol{a}=(x,y)$ ，则 $\left|\boldsymbol{a}\right|^{2}=x^{2}+y^{2}$ ，或 $\left|\boldsymbol{a}\right|=\sqrt{x^{2}+y^{2}}$ .

如果表示向量 $a$ 的有向线段的起点和终点的坐标分别为 $(x_{1},y_{1})$ ， $(x_{2},y_{2})$ ，那么

$\boldsymbol {a} = (x _ {2} - x _ {1}, y _ {2} - y _ {1}),$

$\mid \boldsymbol {a} \mid = \sqrt {(x _ {2} - x _ {1}) ^ {2} + (y _ {2} - y _ {1}) ^ {2}}.$

(2) 设 $\boldsymbol{a}=(x_{1}, y_{1})$ , $\boldsymbol{b}=(x_{2}, y_{2})$ , 则

$\boldsymbol {a} \perp \boldsymbol {b} \Leftrightarrow x _ {1} x _ {2} + y _ {1} y _ {2} = 0.$

设 $a, b$ 都是非零向量，$a = (x_1, y_1)$，$b = (x_2, y_2)$，$\theta$ 是 $a$ 与 $b$ 的夹角，根据向量数量积的定义及坐标表示可得

$\cos \theta = \frac {\boldsymbol {a} \cdot \boldsymbol {b}}{| \boldsymbol {a} | | \boldsymbol {b} |} = \frac {x _ {1} x _ {2} + y _ {1} y _ {2}}{\sqrt {x _ {1} ^ {2} + y _ {1} ^ {2}} \sqrt {x _ {2} ^ {2} + y _ {2} ^ {2}}}.$

公式

对 $\vec{a}=(x_1,y_1)$、$\vec{b}=(x_2,y_2)$：

用法

• 相反向量：$-\vec{b}=(-x_2,-y_2)$（大小相等、方向相反）。
• 把物理/几何向量放进坐标系后，一切加减与求模都化为分量算术。