Question

If $\overrightarrow{A}=(1,1,1),\overrightarrow{C}=(0,1,-1)$ are given vectors, then a vector $\overrightarrow{B}$ satisfying the equations $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{C}$ and $\overrightarrow{A}\cdot \overrightarrow{B}=3$ is $\dots$

Answer 1

Let $\overrightarrow{B}=x\hat{i}+y\hat{j}+z\hat{k}$
Given, $\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{C}=\hat{j}-\hat{k}$
Also, given $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{C}$
$\Rightarrow (z-y)\hat{i}-(z-x)\hat{j}+(y-x)\hat{k}=\hat{j}-\hat{k}$
$\Rightarrow z-y=0,x-z=1,y-x=-1$
$\overrightarrow{A}\cdot \overrightarrow{B}=3\Rightarrow x+y+z=3$
On solving above equations, we get
$x=\frac{5}{3},y=z=\frac{2}{3}$
$\overrightarrow{B}=\left(\frac{5}{3}\hat{i},\frac{2}{3}\hat{j},\frac{2}{3}\hat{k}\right)$