Question

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

• A. $(5,2,2)$
• B. $(\frac{5}{3},\frac{2}{3},\frac{2}{3})$
• C. $(\frac{2}{3},\frac{5}{3},\frac{2}{3})$
• D. $(\frac{2}{3},\frac{2}{3},\frac{5}{3})$

Answer 1

The correct option is B $(\frac{5}{3},\frac{2}{3},\frac{2}{3})$

$\overrightarrow{A}=(1,1,1)\overrightarrow{C}=(0,1,-1)$

Let,

$\overrightarrow{B}=(x,y,z)\overrightarrow{A}\cdot \overrightarrow{B}=3\Rightarrow x+y+z=3\left(1\right)\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{C}(z-y)\hat{i}+(x-z)\hat{j}+(y-x)\hat{k}=\hat{j}-\hat{k}\Rightarrow z=y\left(2\right);x-z=1\left(3\right);y-x=-1\left(4\right)$

Solving $\left(1\right)$ , $\left(2\right)$ , $\left(3\right)$ and $\left(4\right)$ we get:

$x=\frac{5}{3},y=\frac{2}{3},z=\frac{2}{3}$