Question

Given two vectors $\overrightarrow{A}=3\hat{i}+4\hat{j}$ and $\overrightarrow{B}=\hat{i}+\hat{i}$. $\theta$ is the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$. Which of the following statements is/are correct?

• A. $|\overrightarrow{A}|\cos \theta \left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$ is the component of $\overrightarrow{A}$ along $\overrightarrow{B}$.
• B. $|\overrightarrow{A}|\sin \theta \left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$ is the component of $\overrightarrow{A}$ perpendicular to $\overrightarrow{B}$
• C. $|\overrightarrow{A}|\cos \theta \left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)$ is the component of $\overrightarrow{A}$ along $\overrightarrow{B}$.
• D. All of the above

Answer 1

The correct option is A $|\overrightarrow{A}|\cos \theta \left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$ is the component of $\overrightarrow{A}$ along $\overrightarrow{B}$.

$\overrightarrow{A}=3\hat{i}+4\hat{j}$

$\overrightarrow{B}=\hat{i}+\hat{j}$

Unit vector $\hat{B}$ along vector $B$ is $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

A) Component of $\hat{A}$ along $\hat{B}$ $=\left|\overrightarrow{A}\right|cos\theta \left|\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right|$.

B) Unit vector perpendicular to $\overrightarrow{B}=\frac{\hat{i}-\hat{j}}{\sqrt{2}}$