Question

If $a=i+2j+2k,\left|b\right|=5$ and the angle between vector $a$ and $b$ is $\frac{\pi }{6}$, then the area of the triangle formed by these two vectors as two sides is?

• A. $\frac{15}{4}$
• B. $\frac{15}{2}$
• C. $15$
• D. $\frac{15\sqrt{3}}{2}$
• E. $15\sqrt{3}$

Answer 1

The correct option is A

$\frac{15}{4}$

Explanation for the correct option:

Find the area of triangle:

Given, $a=i+2j+2k,\left|b\right|=5$

Now,

$\begin{array}{rcl}\left|a\right|& =& \sqrt{1+2^2+2^2}\\ & =& \sqrt{9}\\ & =& 3\end{array}$

$\therefore$Area $\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}\left|a\times b\right|$

$$\begin{gathered} =\frac{1}{2}\left|a\right|\left|b\right|\sin \theta \\ =\frac{1}{2}\left(3\times 5\times \sin \frac{\mathrm{\pi }}{6}\right) \\ =\frac{1}{2}\left(15\times \frac{1}{2}\right) \\ =\frac{\mathbf{15}}{\mathbf{4}} \end{gathered}$$

Hence, Option ‘A’ is Correct.