Question

Find the vector that must be added to the vector $\hat{i}-3\hat{j}+2\hat{k}$ and $3\hat{i}+6\hat{j}-7\hat{k}$ so that the resultant vector is a unit vector along the $y$- axis.

Answer 1

Given,

$\overrightarrow{A}=\hat{i}-3\hat{j}+2\hat{k}$

$\overrightarrow{B}=3\hat{i}+6\hat{j}-7\hat{k}$

Let $\overrightarrow{C}$ be $x\hat{i}+y\hat{j}+z\hat{k}$

Resultant vector, $R=\hat{j}\left|R\right|=1$

So, $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}=\hat{j}$

$\hat{i}-3\hat{j}+2\hat{k}+3\hat{i}+6\hat{j}-7\hat{k}+x\hat{i}+y\hat{j}+z\hat{k}=\hat{j}$

$(4+x)\hat{i}+2(2+y)\hat{j}+(-5+z)\hat{k}=0$

On comparing, $4+x=0$ $\Rightarrow$ $x=-4$

$2+y=0$ $\Rightarrow$ $y=-2$

$z-5=0$ $\Rightarrow$ $z=5$

$\overrightarrow{C}=-4\hat{i}-2\hat{j}+5\hat{k}$.

It must be added to get the desired result.