Question

$(a.\hat{i})(a\times \hat{i})+(a.\hat{j})(a\times \hat{j})+(.\hat{k})(a\times \hat{k})$ is equal to

• A. 3a
• B. a
• C. 0
• D. None of these

Answer 1

The correct option is C 0

Let $a=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$
Then, $a.\hat{i}=a_1,a.\hat{j}=a_2,a.\hat{k}=a_3$
$a\times \hat{i}=-a_2\hat{k}+a_3\hat{j}.a\times \hat{j}=a_1\hat{k}-a_3\hat{i}$
and $a\times \hat{k}=-a_1\hat{j}+a_2\hat{i}$
Therefore, $(a.\hat{i})(a\times \hat{i})+(a.\hat{j})(a\times \hat{j})+(a.\hat{k})(a\times \hat{k})$
$=a_1(-a_2\hat{k}+a_3\hat{j})+a_2(a_1\hat{k}-a_3\hat{i})+a_3(-a_1\hat{j}+a_2\hat{i})$
$=0$