Question

If $\ell \hat{i}+m \hat{j}+n \hat{k}$ is a unit vector which is perpendicular to vectors $2 \hat{i}-\hat{j}+\hat{k}$ and $3 \hat{i}+4 \hat{j}-\hat{k}$ then find the value of $|\ell|$.

• A. $\frac{3}{\sqrt{155}}$
• B. $\frac{-3}{\sqrt{155}}$
• C. $\frac{5}{\sqrt{155}}$
• D. $\frac{-5}{\sqrt{155}}$

Answer 1

Answer: (A)

Unit vector perpendicular to vectors
$2 \hat{i}-\hat{j}+\hat{k}$ and $3 \hat{i}+4 \hat{j}-\hat{k}$ is $=\frac{(2 \hat{i}-\hat{j}+\hat{k}) \times(3 \hat{i}+4 \hat{j}-\hat{k})}{|(2 \hat{i}-\hat{j}+\hat{k}) \times(3 \hat{i}+4 \hat{j}-\hat{k})|}$
$=\frac{\hat{i}(1-4)-\hat{j}(-2-3)+\hat{k}(8+3)}{\sqrt{9+25+121}}=\frac{-3 \hat{i}+5 \hat{j}+11 \hat{k}}{\sqrt{155}}$
$\therefore |\ell|=\left|\frac{-3}{\sqrt{155}}\right|=\frac{3}{\sqrt{155}}$