Question

The direction cosines of $\overline{AB}$ are $-2,2,1$. If $A\equiv (4,1,5)$ and $l\left(AB\right)=6$units, find the coordinates of $B$.

Answer 1

Given direction ratio of $AB=-2,2,1$
Note= if any line have direction ratio =$(a,b,c)$ then $(ka,kb,kc)$ is also the direction ratio of that line..
$A=(x_1,y_1,z_1)=(4,1,5)$
Let $B=(x_2,y_2,z_2)$
hence $(x_2-x_1)=-2K,(y_2-y_1)=2K,(z_2-z_1)=K$ .........equ($1$)
hence $x_2=-2\times K+4$
$y_2=2\times K+1$
$z_2=K+5$
$l\left(AB\right)=6$ units
Apply distance formula $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$ ....... equ($2$)
using equation($1$) and equ($2$)
$6=\sqrt{(-2\times K{)}^2+(2\times K{)}^2+K^2}$
on solving this equation, we will get
$9\times K^2=36$
$K=2$ or $-2$
Now using equation($1$)
For $K=2$ $B=(0,5,7)$
And for $K=-2$ $B=(8,-3,3)$