Question

A unit vector perpendicular to each of the vector $2\mathbf{i} - \mathbf{j} + \mathbf{k}$ and $3\mathbf{i} + 4\mathbf{j} - \mathbf{k}$ is equal to

• A. $\frac{( - 3\mathbf{i} + 5\mathbf{j} + 11\mathbf{k})}{\sqrt{155}}$
• B. $\frac{(3\mathbf{i} - 5\mathbf{j} + 11\mathbf{k})}{\sqrt{155}}$
• C. $\frac{(6\mathbf{i} - 4\mathbf{j} - \mathbf{k})}{\sqrt{53}}$
• D. $\frac{(5\mathbf{i} + 3\mathbf{j})}{\sqrt{34}}$

Answer 1

Answer: (A)

$\frac{( - 3\mathbf{i} + 5\mathbf{j} + 11\mathbf{k})}{\sqrt{155}}$

Answer 2

Let $\mathbf{a} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}$ and $\mathbf{b} = 3\mathbf{i} + 4\mathbf{j} - \mathbf{k},$ then a unit vector perpendicular to $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\mathbf{a} \times \mathbf{b}}{|\mathbf{a} \times \mathbf{b}|}$

Here $\mathbf{a} \times \mathbf{b} = - 3\mathbf{i} + 5\mathbf{j} + 11\mathbf{k}$

Unit vector is $\frac{- 3\mathbf{i} + 5\mathbf{j} + 11\mathbf{k}}{\sqrt{155}}$.