Question

Find the angle between \(\vec{a} \,and \,\vec{b}\) if \(|\vec{a}|=2,|\vec{b}|=\frac{1}{2√3}\) and \(\vec{a}×\vec{b}=\frac{1}{2}\).

(a) \(\frac{2π}{3}\)

(b) \(\frac{4π}{5}\)

(c) \(\frac{π}{3}\)

(d) \(\frac{π}{2}\)

I got this question during an interview for a job.

My doubt is from Product of Two Vectors-2 topic in chapter Vector Algebra of Mathematics – Class 12

Answer 1

The correct option is (c) \(\frac{π}{3}\)

To explain: Given that, \(|\vec{a}|=2, \,|\vec{b}|=\frac{1}{2√3}\) and \(\vec{a}×\vec{b}=\frac{1}{2}\)

We know that, \(\vec{a}×\vec{b}=\vec{a}.\vec{b} \,sin⁡θ\)

∴ \(sin⁡θ=\frac{(\vec{a}×\vec{b})}{|\vec{a}|.|\vec{b}|}\)

sin⁡θ=\(\frac{\frac{1}{2}}{2×\frac{1}{2√3}}=\frac{\sqrt{3}}{2}\)

θ=\(sin^{-1}⁡\frac{\sqrt{3}}{2}=\frac{π}{3}\)