Question

If $a$ and $b$ are unit vectors, then the vector $(a+b)\times (a\times b)$ is parallel to the vector?

• A. $a–b$
• B. $a+b$
• C. $2a–b$
• D. $2a+b$

Answer 1

The correct option is A

$a–b$

Explanation for the correct option:

Given, $(a+b)\times (a\times b)$

$=a\times (a\times b)+b\times (a\times b)$

$$\begin{gathered} =(a·b)a-(a·a)b+(b·b)a-(b·a)b \\ =(a·b)(a-b)+a-b \\ =(a·b+1)(a-b) \end{gathered}$$ $\left[\because a·a=1,b·b=1\right]$

$\Rightarrow$ $(a+b)\times (a\times b)=k(a-b)$ where $k=a·b+1$ is a scalar

$\mathbf{\therefore }\mathbf{(}\mathit{a}\mathbf{}\mathbf{+}\mathbf{}\mathit{b}\mathbf{)}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{(}\mathit{a}\mathbf{}\mathbf{\times }\mathbf{}\mathit{b}\mathbf{)}\mathbf{}\mathbf{\parallel }\left(a-b\right)$

Hence, Option ‘A’ is Correct.