Question

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

Answer 1

Let the coordinates of point B be $\left(x_1,y_1,z_1\right)$

direction ratios of AB are $-2,2,1$

So, $x_1-4=-2r$

$y_1-1=2r$

$z_1-5=r$

But lenght of AB $=6$

$\Rightarrow \sqrt{{\left(x_1-4\right)}^2+{\left(y_1-1\right)}^2+{\left(z_1-5\right)}^2}=6$

$\Rightarrow {\left(x_1-4\right)}^2+{\left(y_1-1\right)}^2+{\left(z_1-5\right)}^2=36$

$\Rightarrow 4r^2+4r^2+r^2=36$

$\Rightarrow 9r^2=36$

$\Rightarrow r=2$ or $r=-2$

if $r=2$

$x_1=0,$$y_1=5,$$z_1=7$

and if $r=-2$

$x_1=8,$$y_1=-3,$$z_1=3$

Therefore the point B is $\left(0,5,7\right)$ or $\left(8,-3,3\right)$