Question

The projection of line joining $(3,4,5)$ and $(4,6,3)$ on the line joining $(-1,2,4)$ and $(1,0,5)$ is

• A. $\frac{4}{3}$
• B. $\frac{2}{3}$
• C. $\frac{8}{3}$
• D. $\frac{1}{3}$

Answer 1

Answer: (A)

Let $A=(3,4,5),P=(-1,2,4)$
$B=(4,6,3)$ and $Q=(1,0,5)$
$\therefore$ Dr's of line $AB$ are $(4-3),(6-4),(3-5)=1,2,-2$
and Dr's of line $PQ$ are $(1+1),(0-2),(5-4)=2,-2,1$
$\therefore$ Dc's of line $PQ=\frac{2}{\sqrt{2^2+(-2{)}^2+1}},\frac{-2}{3},\frac{1}{3}$
$=\frac{2}{3},\frac{-2}{3},\frac{1}{3}$
$\therefore$ Projection of line segment $AB$ on the line $PQ$ is
$\left|\frac{2}{3}(1)+\left(\frac{-2}{3}\right)(2)+\left(\frac{1}{3}\right)(-2)\right|=\frac{4}{3}$