Question

If $\left|a\right|=10,\left|b\right|=2anda.b=12,$ then $|a\times b|$ is equal to?

• A. $12$
• B. $14$
• C. $16$
• D. $18$

Answer 1

The correct option is C

$16$

Explanation for the correct option:

Step1. Finding value of $\sin \theta \&\cos \theta$

Given, $\left|a\right|=10,\left|b\right|=2anda.b=12,$

$$\begin{gathered} Weknowthat, \\ \mathit{a}\mathbf{.}\mathit{b}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathit{a}\mathbf{\right|}\mathbf{}\mathbf{\left|}\mathit{b}\mathbf{\right|}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta } \\ Cos\theta =\frac{(a.b)}{\left(\right|a|.|b\left|\right)} \\ Cos\theta =\frac{12}{(10x2)} \\ Cos\theta =\frac{3}{5} \end{gathered}$$

Then,

$$\begin{gathered} Sin\theta ={(1-{\cos }^2\theta )}^{\frac{1}{2}} \\ Sin\theta ={\left(1-{\left(\frac{3}{5}\right)}^2\right)}^{\frac{1}{2}} \\ Sin\theta =\frac{4}{5} \end{gathered}$$

Step 2. Finding value of vector product.

|$$\begin{gathered} \Rightarrow \mathbf{|}\mathit{a}\mathbf{}\mathit{x}\mathbf{}\mathit{b}\mathbf{|}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathit{a}\mathbf{\right|}\mathbf{}\mathbf{\left|}\mathit{b}\mathbf{\right|}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta } \\ \Rightarrow |axb|=10\times 2\times \frac{4}{5} \\ \therefore |axb|=16 \end{gathered}$$

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