Question

Find the direction consists of the vector joining the point $A(1,2,-3)$ and $B(-1,-2,1)$, directed from $A$ to $B$.

Answer 1

$A(1,2,-3)$; $B(-1,-2,1)$

$\overline{OA}=\hat{i}+2\hat{j}-3\hat{k}$ ,

$\overline{OB}=-\hat{i}-2\hat{j}+\hat{k}$

$\overline{AB}=\overline{OB}-\overline{OA}=(-\hat{i}-2\hat{j}+\hat{k})-(\hat{i}+2\hat{j}-3\hat{k})$

$=-2\hat{i}-4\hat{j}+4\hat{k}$

D.C of vector $-2\hat{i}-4\hat{j}+4\hat{k}$ is

$(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}})$

$=(\frac{-2}{\sqrt{4+16+16}},\frac{-4}{\sqrt{4+16+16}},\frac{4}{\sqrt{4+16+16}})=(\frac{-2}{\sqrt{36}},\frac{-4}{\sqrt{36}},\frac{4}{\sqrt{36}})=(\frac{-2}{6},\frac{-4}{6},\frac{4}{6})=(\frac{-1}{3},\frac{-2}{3},\frac{2}{3})$