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Question: Let \(\mathbf { a } , \mathbf { b }\) and<img src="https://cdn.pureessence.tech/canvas_119.png?top_l...

Let a,b\mathbf { a } , \mathbf { b } and be non-zero vectors such that . If θ\theta is the acute angle between the vectors and , then sin⁡θ\sin \theta equals

A

223\frac { 2 \sqrt { 2 } } { 3 }

B

23\frac { \sqrt { 2 } } { 3 }

C

23\frac { 2 } { 3 }

D

13\frac { 1 } { 3 }

Answer

223\frac { 2 \sqrt { 2 } } { 3 }

Explanation

Solution

∙∙\bullet \bullet (a×b)×c( \mathbf { a } \times \mathbf { b } ) \times \mathbf { c } = 13∣b∥c∣a\frac { 1 } { 3 } | \mathbf { b } \| \mathbf { c } | \mathbf { a }

̃

̃

̃ As a\mathbf { a } and are not parallel, a. c=0\mathbf { c } = 0 and cos⁡θ+13=0\cos \theta + \frac { 1 } { 3 } = 0

̃ cos⁡θ=−13\cos \theta = - \frac { 1 } { 3 } ̃ sin⁡θ=223\sin \theta = \frac { 2 \sqrt { 2 } } { 3 }