Question

Let `veca` be any vector such that `|veca| = a`. The value of `|vecaxxhati|^2 + |vecaxxhatj|^2 + |vecaxxhatk|^2` is ______.

Options

• a²

• 2a²

• 3a²

• 0

Answer 1

Let `veca` be any vector such that `|veca| = a`. The value of `|vecaxxhati|^2 + |vecaxxhatj|^2 + |vecaxxhatk|^2` is 2a².

Explanation:

Let `veca = a_1hati + a_2hatj + a_3hatk`.

Given `|veca| = a`

Then `vecaxxhati = -a_2hatk+a_3hatj  ...[therefore hatixxhati = hatjxxhati = hatkxxhatk = 0]` 

`vecaxxhatj = a_1hatk-a_3hatj`

`vecaxxhatk=-a_1hatj + a_2hati`

and `therefore |vecaxxhati|^2+|vecaxxhatj|^2+|vecaxxhatk|^2  ...[therefore |veca| = root  (a_1^2 + a_2^2 + a_3^2)]`

`= a_2^2 + a_3^2 + a_1^2 + a_3^2+a_1^2 + a_2^2`

`= 2[a_1^2+a_2^2 + a_3^2]`

`= 2[|veca|]^2`

= 2a²