Question

The angle between two vectors given by $6\bar{i} + 6\bar{j} - 3\bar{k}$ and $7\overline{i} + 4\overline{j} + 4\overline{k}$ is

• A. $\cos^{- 1}\left( \frac{1}{\sqrt{3}} \right)$
• B. $\cos^{- 1}\left( \frac{5}{\sqrt{3}} \right)$
• C. $\sin^{- 1}\left( \frac{2}{\sqrt{3}} \right)$
• D. $\sin^{- 1}\left( \frac{\sqrt{5}}{3} \right)$

Answer 1

Answer: (D)

$\sin^{- 1}\left( \frac{\sqrt{5}}{3} \right)$

Answer 2

$\cos\theta = \frac{\overrightarrow{A}\overrightarrow{B}}{AB} = \frac{42 + 24 - 12}{\sqrt{36 + 36 + 9}\sqrt{49 + 16 + 16}} = \frac{56}{9\sqrt{71}}$

$\cos\theta = \frac{56}{9\sqrt{71}}$ ∴ $\sin\theta = \frac{\sqrt{5}}{3}$ or $\theta = \sin^{- 1}\left( \frac{\sqrt{5}}{3} \right)$