Question

For any vector $\vec{a}$
$\hat{i} \times (\vec{a} \times \hat{i} ) + \hat{j} \times (\vec{a} \times \hat{j} ) + \hat{k} \times (\vec{a} \times \hat{k} )$

• A. $\hat{i} + \hat{j} + \hat{k}$
• B. $3 \vec{a}$
• C. $ \vec{a}$
• D. $2 \vec{a}$

Answer 1

Answer: (D)

$\therefore \:\:\:\: \vec{a} \times \hat{i} = x (\hat{i} \times \hat{i} ) + y (\hat{j} \times \hat{i} ) + z(\hat{k} \times \hat{i} ) = - y \hat{k} + z \hat{j} $
Similarly, $\vec{a} \times \hat{j} = x (\hat{i} \times \hat{j} ) + y (\hat{j} \times \hat{j} ) + z(\hat{k} \times \hat{j} )$
          $ = x\hat{k} -z \hat{i} $
and $\vec{a} \times \hat{k} = x (\hat{i} \times \hat{k} ) + y (\hat{j} \times \hat{k} ) + z(\hat{k} \times \hat{k} ) $
          $= -x \hat{j} + y \hat{i} $
Now , $ \hat{i} \times (\vec{a} \times \hat{i} ) + \hat{j} \times (\vec{a} \times \hat{j} ) + \hat{k} \times (\vec{a} \times \hat{k} ) $
     $ = \hat{i} \times ( - y\hat{k} \times z\hat{j} ) + \hat{j} \times ( - z\hat{i} \times x\hat{k} ) + \hat{k} \times ( - x\hat{j} \times y \hat{i})$
     $ = y\hat{j} + z\hat{k} +z\hat{k} + x\hat{i} + x\hat{i}+ y \hat{j}$
      $= 2 ( x\hat{i} + y\hat{j} + z \hat{k} ) = 2 \vec{a} $