Question

A unit vector parallel to $xy$-plane and perpendicular to the vector $4\hat{i}-3\hat{j}+\hat{k}$ is

• A. $\pm \frac{1}{5}(3\hat{i}+4\hat{k})$
• B. $\pm \frac{1}{5}(4\hat{i}+3\hat{j})$
• C. $\pm \frac{1}{5}(3\hat{i}+4\hat{j})$
• D. $\hat{i}+2\hat{k}$

Answer 1

Answer: (C)

$\pm \frac{1}{5}(3\hat{i}+4\hat{j})$

The vector parallel to $xy$ plane is perpendicular to $\hat{k}$ in the z-direction

$\therefore \hat{k}\times (4\hat{i}-3\hat{j}+\hat{k})=(4\hat{j}+3\hat{i})$ is the vector in the $x-y$ plane perpendicular to both $\hat{k}$ and the given vector

$\therefore$ the required unit vector

$=\frac{3\hat{i}+4\hat{j}}{|3\hat{i}+4\hat{j}|}=\frac{3\hat{i}+4\hat{j}}{\sqrt{3^2+4^2}}=\pm \frac{1}{5}(3\hat{i}+4\hat{j})$