Question

If $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{u}$ is a vector satisfying $\overrightarrow{a}\times \overrightarrow{u}=\overrightarrow{a}\times \overrightarrow{b}$ and $\overrightarrow{a}\cdot \overrightarrow{u}=0$, then $2{\left|\overrightarrow{u}\right|}^2$ is equal to

• A. 5
• B. 4
• C. 8
• D. None of these

Answer 1

The correct option is A 5

$a\times u=a\times b\Rightarrow a\times (u-b)=0$
$\Rightarrow u-b=ta$ for some $t$
$\Rightarrow u=b+ta\Rightarrow u=(2+t)i+(1+2t)j-(1+3t)k$
Now $a\cdot u=0\Rightarrow 2+t+2(1+2t)+3(1+3t)=0$
$\Rightarrow t=-1/2.$ Hence $u=\left(3/2\right)i+\left(1/2\right)k$ and $2{\left|u\right|}^2=5.$