Question

Show that (×)× = ×(×) if and only if the vectors and are collinear.

Answer 1

Explanation:

NOTE: Bolded letters are vectors

Given that ,

• (axb)xc = ax(bxc)

From vector triple product rule,

• ax(bxc) = b(a.c)-c(a.b)

• ax(bxc) = b(a.c)-a(b.c)

If these both are equal, then :

⇒b(a.c)-c(a.b) = b(a.c)-a(b.c)

⇒(-a.b)c = (-b.c)a

⇒(|a||b|sin∅)c = (|b||c|sinФ)a (cancel |b| on both sides)

A , C are the unit vectors of a and c respectively .

Two vectors can be equal on when their magnitude and directions are same.

That is when ,

⇒(|a||c|sin∅)A = (|a||c|sinФ)C

⇒sin∅A = sinФC

This can only be possible when the vectors are collinear because the angle made by the vectors with the vector B should be the same or supplementary and the vectors A and C must be in the same direction.