Question

If \(\vec{a} \,and \,\vec{b}\) are two non-zero vectors then \((\vec{a}-\vec{b})×(\vec{a}+\vec{b})\)=_________

(a) \(2(\vec{a}×\vec{b})\)

(b) \((\vec{a}×\vec{b})\)

(c) –\(4(\vec{a}×\vec{b})\)

(d) \(3(\vec{a}×\vec{b})\)

This question was posed to me during an online interview.

My doubt stems from Product of Two Vectors-2 in portion Vector Algebra of Mathematics – Class 12

Answer 1

The correct option is (a) \(2(\vec{a}×\vec{b})\)

To explain: Consider \((\vec{a}-\vec{b})×(\vec{a}+\vec{b})\)

=\((\vec{a}-\vec{b})×\vec{a}+(\vec{a}+\vec{b})×\vec{b}\)

=\(\vec{a}×\vec{a}-\vec{b}×\vec{a}+\vec{a}×\vec{b}-\vec{b}×\vec{b}\)

We know that, \(\vec{a}×\vec{a}=0,\vec{b}×\vec{b}=0 \,and \,\vec{a}×\vec{b}=-\vec{b}×\vec{a}\)

∴ \(\vec{a}×\vec{a}-\vec{b}×\vec{a}+\vec{a}×\vec{b}-\vec{b}×\vec{b}=0+2(\vec{a}×\vec{b})+0\)

Hence, \((\vec{a}-\vec{b})×(\vec{a}+\vec{b})=2(\vec{a}×\vec{b})\)