Question

If $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{b}=\hat{i}\times (\overrightarrow{a}\times \hat{i})+\hat{j}\times (\overrightarrow{a}\times \hat{j})+\overrightarrow{k}\times (\overrightarrow{a}\times \hat{k})$ then length of $\overrightarrow{b}$ isequal to

• (A) √12
• (B) 2√12
• (C) 3√14
• (D) 2√14

Answer 1

The Correct Option is D

Explanation:
$\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+2\hat{\mathbf{j}}+3\hat{\mathbf{k}}$ and$\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}\times (\overrightarrow{\mathbf{a}}\times \hat{\mathbf{i}})+\hat{\mathbf{j}}\times (\overrightarrow{\mathbf{a}}\times \hat{\mathbf{j}})+\hat{\mathbf{k}}\times (\overrightarrow{\mathbf{a}}\times \hat{\mathbf{k}})$$=3\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{a}}=2\overrightarrow{\mathbf{a}}$$=2(\hat{\mathbf{i}}+2\hat{\mathbf{j}}+3\hat{\mathbf{k}})$$\Rightarrow \overrightarrow{\mathbf{b}}\mid =\sqrt{4+16+36}$$=\sqrt{56}$$=2\sqrt{14}$