Question

Find the direction cosines of the vector joining the points $A(1,2,-3)andB(-1,-2,1)$, directed from A to B.

Answer 1

The given points are $A(1,2,-3)andB(-1,-2,1)$.
$i.e.,x_1=1,y_1=2,z_1=-3and{x}_2=-1,y_2=-2,z_2=1$
Vector $AB=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}=(-1-1)\hat{i}+(-2-2)\hat{j}+[1-(-3\left)\right]\hat{k}=-2\hat{i}-4\hat{j}+4\hat{k}$
Comparing with $X=x\hat{i}+y\hat{j}+z\hat{k}wegetx=-2,y=-4,z=4Now,magnitude$
$|AB|=\sqrt{x^2+y^2+z^2}=\sqrt{(-2{)}^2+(-4{)}^2+4^2}=\sqrt{4+16+16}=\sqrt{36}=6$
Direction cosines of a vector $X=x\hat{i}+y\hat{j}+z\hat{k}are\frac{x}{|X|},\frac{y}{|X|},\frac{z}{|X|}$
$\therefore$ Direction cosines of AB are $\frac{-2}{6},\frac{-4}{6},\frac{4}{6}or\frac{-1}{3},\frac{-2}{3},\frac{2}{3}.$