Question

If $\hat{a}$ is a unit vector and projection of $\overrightarrow{x}$ along $\hat{a}$ is $2$ units and $(\hat{a}\times \overrightarrow{x})+\overrightarrow{b}=\overrightarrow{x}$, then $\overrightarrow{x}$ is equal to

• A. $\frac{1}{2}[\hat{a}-\overrightarrow{b}+(\hat{a}\times \overrightarrow{b}\left)\right]$
• B. $\frac{1}{2}[2\hat{a}+\overrightarrow{b}+(\hat{a}\times \overrightarrow{b}\left)\right]$
• C. $[\hat{a}+(\hat{a}\times \overrightarrow{b}\left)\right]$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{2}[2\hat{a}+\overrightarrow{b}+(\hat{a}\times \overrightarrow{b}\left)\right]$

$(\hat{a}\times x)+b=x$
$\Rightarrow \hat{a}\times (\hat{a}\times x)+(\hat{a}\times b)=\hat{a}\times x$
$(\hat{a}\cdot x)\hat{a}-(\hat{a}\cdot \hat{a})x+(\hat{a}\times b)=x-b$

Projection of x along $\hat{a}$ is $2$ units
$\Rightarrow \frac{(\hat{a}\cdot x)}{|\hat{a}|}=2$
$\Rightarrow \hat{a}\cdot x=2$

So, $x=\frac{1}{2}[2\hat{a}+b+(\hat{a}\times b\left)\right]$