Question

If vectors $A=(1,1,1),C=(0,1,-1)$ are given and $B$ is a vector satisfying the equations $A\times B=C$ and $A\cdot B=3$, then find the value of $9{\left|B\right|}^2$.

• A. 9
• B. 81
• C. 33
• D. 27

Answer 1

Answer: (C)

33

Let $B=x\hat{i}+y\hat{j}+z\hat{k}$

Given, $A=\hat{i}+\hat{j}+\hat{k}$

Also given, $A\times B=C$$\Rightarrow (z-y)\hat{i}-(z-x)\hat{j}+(y-x)\hat{k}=\hat{j}-\hat{k}$

$\Rightarrow z-y=0,x-z=1,y-x=-1$

Also, $A.B=3$$\Rightarrow x+y+z=3$

On solving the equations we get,

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

$\therefore B(\frac{5}{3},\frac{2}{3},\frac{2}{3})$

Then $9B^2=33$