Question

If p, q and r are non-zero, non-coplanar vectors then [p + q - r  p - q  q - r] = ______.

Options

• 3[p q r]

• 0

• [p q r]

• 2[p q r]

Answer 1

If p, q and r are non-zero, non-coplanar vectors then [p + q - r  p - q  q - r] = [p q r].

Explanation:

Given, p, q and r are non-zero coplanar vectors.

∴ [p + q - r  p - q  q - r]

= (p + q - r) · [(p - q) × (q - r)]

= (p + q + r) · [p × q - p × r - q × q + q × r]

= (p + q - r) . [p × q - p × r + q × r]  (∵ q × q = 0)

= p . (p × q) - p . (p × r) + p . (q × r) + q . (p × q) - q . (p × r) + q . (q × r) - r . (p × q) - r . (p × r) + r . (q × r)

= [p q r] - [q p r] - [r p q]

= [p q r] + [p q r] - [p q r]

= 2[p q r] - [p q r] = [p q r]