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Question: If \(|\mathbf{a}.\mathbf{b}| = 3\) and \(|\mathbf{a} \times \mathbf{b}| = 4,\) then the angle betwee...

If ∣a.b∣=3|\mathbf{a}.\mathbf{b}| = 3 and ∣a×b∣=4,|\mathbf{a} \times \mathbf{b}| = 4, then the angle between a and b is

A

cos⁡−134\cos^{- 1}\frac{3}{4}

B

cos⁡−135\cos^{- 1}\frac{3}{5}

C

cos⁡−145\cos^{- 1}\frac{4}{5}

D

π4\frac{\pi}{4}

Answer

cos⁡−135\cos^{- 1}\frac{3}{5}

Explanation

Solution

∣a.b∣=abcos⁡θ=3|\mathbf{a}.\mathbf{b}| = ab\cos\theta = 3 …..(i)

and ∣a×b∣=absin⁡θ=4|\mathbf{a} \times \mathbf{b}| = ab\sin\theta = 4 …..(ii)

Dividing (ii) by (i),

we get tan⁡θ=43⇒cos⁡θ=35⇒θ=cos⁡−135.\tan\theta = \frac{4}{3} \Rightarrow \cos\theta = \frac{3}{5} \Rightarrow \theta = \cos^{- 1}\frac{3}{5}.