例题 1.1.

例题 1.1.

求点 $P\left(1,2,0\right)$ 和点 $Q\left(2,1,\sqrt{2}\right)$ 的距离 $d\left(P,Q\right)$ 、向量 $\overrightarrow{PQ}$ 的长度、方向余弦及方向角.

解

$$\begin{array}{rl}\overrightarrow{PQ}& =(2-1)\mathbf{i}+(1-2)\mathbf{j}+(\sqrt{2}-0)\mathbf{k}\\ & =\mathbf{i}-\mathbf{j}+\sqrt{2}\mathbf{k}.\end{array}$$ $$d\left(P,Q\right)=\left|\overrightarrow{PQ}\right|=\sqrt{1^2+{\left(-1\right)}^2+{\left(\sqrt{2}\right)}^2}=2.$$ $$\cos \alpha =\frac{1}{2},\cos \beta =-\frac{1}{2},\cos \gamma =\frac{\sqrt{2}}{2}.$$ $$\alpha =\frac{\pi }{3},\beta =\frac{2\pi }{3},\gamma =\frac{\pi }{4}.$$