Question

If $a\&b$ are any two vectors, then $(2a+3b)\times (5a+7b)+a\times b$ is equal to?

• A. $0$
• B. $1$
• C. $a\times b$
• D. $b\times a$

Answer 1

The correct option is A

$0$

Explanation for the correct option:

Finding the value:

Given, $a\&b$ are two vectors

Now,

$\begin{array}{rcl}& & (2a+3b)\times (5a+7b)+a\times b\\ & =& 2ax(5a+7b)+3bx(5a+7b)+axb\\ & =& 10\left(axa\right)+14\left(axb\right)+15\left(bxa\right)+21\left(bxb\right)+axb\\ & =& 14axb–15\left(axb\right)+axb\mathbf{}\mathbf{[}\mathbf{\left(}\mathbf{a}\mathbf{x}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{\&}\mathbf{}\mathbf{\left(}\mathbf{b}\mathbf{x}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathbf{(}\mathit{b}\mathit{x}\mathit{a}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{(}\mathit{a}\mathit{x}\mathit{b}\mathbf{\left)}\mathbf{\right]}\\ & & \\ & =& 0\\ & & \end{array}$

Hence, correct option is $\mathbf{\left(}\mathit{A}\mathbf{\right)}$