Question

If $\left|a\right|=1$, $\left|b\right|=4,a.b=2$ and $c=2a\times b–3b$, then the angle between $b$ and $c$ is?

• A. $\frac{\mathrm{\pi }}{6}$
• B. $\frac{5\mathrm{\pi }}{6}$
• C. $\frac{\mathrm{\pi }}{3}$
• D. $\frac{2\mathrm{\pi }}{3}$

Answer 1

The correct option is B

$\frac{5\mathrm{\pi }}{6}$

Explanation for the correct option:

Step 1. Find the value of $\mathbf{\left|}\mathit{c}\mathbf{\right|}$:

Given, $\left|a\right|=1,\left|b\right|=4,a.b=2$ and $c=2a\times b–3b$

We now that $|a\times b{|}^2=|a{|}^2|b{|}^2-{(a.b)}^2$

$\Rightarrow$ $\begin{array}{rcl}|a\times b{|}^2& =& 16-4\\ & =& 12\end{array}$

And given that $c=2a\times b–3b$

$\Rightarrow$ $|c{|}^2={(2a\times b-3b)}^2$

$\Rightarrow$ $|c{|}^2=4|a\times b{|}^2+9|b{|}^2$

$\Rightarrow$ $|c{|}^2=4(12)+9(16)$

$\Rightarrow$ $|c{|}^2=192$

$\Rightarrow$ $\left|c\right|=8\sqrt{3}$

Step 2. Find the angle between $b$ and $c$ :

We have $b.c=b(2a\times b-3b)$

$\Rightarrow$ $b.c=-3|b{|}^2$

$\Rightarrow$ $b.c=-48$

The angle between $b$ and $c$ is given by $\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\mathbf{}\mathbf{=}\frac{\mathbf{(}\mathbf{b}\mathbf{.}\mathbf{c}\mathbf{)}}{\mathbf{\left(}\mathbf{\right|}\mathbf{b}\mathbf{\left|}\mathbf{\right|}\mathbf{c}\mathbf{\left|}\mathbf{\right)}}\mathbf{}$

$$\begin{gathered} \Rightarrow \cos \theta =\frac{-48}{(4\times 8\sqrt{3})} \\ \Rightarrow \cos \theta =\frac{-\sqrt{3}}{2} \\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{-\sqrt{3}}{2}\right) \\ \Rightarrow \theta =\frac{\mathbf{}\mathbf{5}\mathbf{\pi }}{\mathbf{6}}\mathbf{} \end{gathered}$$

Hence, the correct option is (B).