Question

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

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

Answer 1

The correct option is A $\frac{4}{3}$

Let the $4$ points be $A,B,C,D$
So, $A(3,4,5),B(4,6,3),C(-1,2,4),D(1,0,5)$
Vectors are:
$\overrightarrow{AB}=(4-3)\hat{i}+(6-4)\hat{j}+(3-5)\hat{k}=\hat{i}+2\hat{j}-2\hat{k}$
$\overrightarrow{CD}=(1+1)\hat{i}+(0-2)\hat{j}+(5-4)\hat{k}=2\hat{i}-2\hat{j}+\hat{k}$
Projection of $\overrightarrow{AB}$ on $\overrightarrow{CD}$ is $=\left|\frac{\overrightarrow{AB}\cdot \overrightarrow{CD}}{\left|\overrightarrow{CD}\right|}\right|=\left|\frac{(\hat{i}+2\hat{j}-2\hat{k})\cdot (2\hat{i}-2\hat{j}+\hat{k})}{\sqrt{4+4+1}}\right|=\left|\frac{2-4-2}{3}\right|=\frac{4}{3}$