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

Step-by-step 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.