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Question: If \(|\mathbf{a}| = 2,|\mathbf{b}| = 5\) and \(|\mathbf{a} \times \mathbf{b}| = 8,\) then **a . b** ...

If ∣a∣=2,∣b∣=5|\mathbf{a}| = 2,|\mathbf{b}| = 5 and ∣a×b∣=8,|\mathbf{a} \times \mathbf{b}| = 8, then a . b is equal to

A

0

B

2

C

4

D

6

Answer

6

Explanation

Solution

a.b=∣a∣∣b∣cos⁡θ\mathbf{a}.\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta; But ∣a×b∣=∣a∣∣b∣sin⁡θ|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta

⇒sin⁡θ=∣a×b∣∣a∣∣b∣=45⇒cos⁡θ=35\Rightarrow \sin\theta = \frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}||\mathbf{b}|} = \frac{4}{5} \Rightarrow \cos\theta = \frac{3}{5}

Therefore, a.b=2×5×35=6.\mathbf{a}.\mathbf{b} = 2 \times 5 \times \frac{3}{5} = 6.