Question

$\hat{i}and\hat{j}$ are unit vectors along along x- and y- axis respectively. What is the magnitude and direction of the vectors $\hat{i}+\hat{j}$.and $\hat{i}-\hat{j}$ ?What are the components of a vector A= 2$\hat{i}+3\hat{j}$ along the directions of $\hat{i}+\hat{j}$ and $\hat{i}-\hat{j}$ ?[You may use graphical method]

Answer 1

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

$|\overrightarrow{A}|=\sqrt{1^2+1^2}=\sqrt{2}$

Angle of $\overrightarrow{A}$ with x-axis i.e. $\hat{i},{\theta }_A={tan}^{-1}\left(\frac{1}{1}\right)=\pi /4$

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

$|\overrightarrow{B}|=\sqrt{1^2+(-1{)}^2}=\sqrt{2}$

Angle of $\overrightarrow{B}$ with x-axis i.e. $\hat{j},{\theta }_B={tan}^{-1}\left(\frac{-1}{1}\right)=-\pi /4$

$\overrightarrow{C}=2\hat{i}+3\hat{j}$

Component of $\overrightarrow{C}$ along $\overrightarrow{A},x_A=\frac{\overrightarrow{A}.\overrightarrow{C}}{|\overrightarrow{A}|}$

$x_A=\frac{5}{\sqrt{2}}$

Component of $\overrightarrow{C}$ along $\overrightarrow{B},x_B=\frac{\overrightarrow{B}.\overrightarrow{C}}{|\overrightarrow{B}|}$

$x_B=\frac{-1}{\sqrt{2}}$