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Question: The angle between the vectors \(3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) and \(2\mathbf{i} - 2\mathb...

The angle between the vectors 3i+j+2k3\mathbf{i} + \mathbf{j} + 2\mathbf{k} and 2i−2j+4k2\mathbf{i} - 2\mathbf{j} + 4\mathbf{k} is

A

cos⁡−127\cos^{- 1}\frac{2}{\sqrt{7}}

B

sin⁡−127\sin^{- 1}\frac{2}{\sqrt{7}}

C

cos⁡−125\cos^{- 1}\frac{2}{\sqrt{5}}

D

sin⁡−125\sin^{- 1}\frac{2}{\sqrt{5}}

Answer

sin⁡−127\sin^{- 1}\frac{2}{\sqrt{7}}

Explanation

Solution

cos⁡θ=3(2)+(1)(−2)+2(4)9+1+44+4+16=121424=6146\cos\theta = \frac{3(2) + (1)( - 2) + 2(4)}{\sqrt{9 + 1 + 4}\sqrt{4 + 4 + 16}} = \frac{12}{\sqrt{14}\sqrt{24}} = \frac{6}{\sqrt{14}\sqrt{6}}

⇒cos⁡θ=37⇒sin⁡θ=27\Rightarrow \cos\theta = \frac{\sqrt{3}}{\sqrt{7}} \Rightarrow \sin\theta = \frac{2}{\sqrt{7}} ⇒ θ=sin⁡−1(27)\theta = \sin^{- 1}\left( \frac{2}{\sqrt{7}} \right).