Question

Find the components of a vector $\overrightarrow{A}=2\hat{i}+3\hat{j}$ along the directions of $\hat{i}+\hat{j}and\hat{i}-\hat{j}.$

• A. $\frac{5}{\sqrt{2}},\frac{-1}{\sqrt{2}}$
• B. $\frac{-5}{\sqrt{2}},\frac{-1}{\sqrt{2}}$
• C. $\frac{5}{\sqrt{2}},\frac{1}{\sqrt{2}}$
• D. $\frac{-5}{\sqrt{2}},\frac{1}{\sqrt{2}}$

Answer 1

The correct option is A $\frac{5}{\sqrt{2}},\frac{-1}{\sqrt{2}}$

Step 1: Find the unit vectvectors in the direction of the given vectors
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be the unit vectors in the direction of the given vectors

$\hat{i}+\hat{j}$ and $\hat{i}-\hat{j}$ respectively.

$\overrightarrow{a}=\frac{\hat{i}+\hat{j}}{|\hat{i}+\hat{j}|}$

$|\hat{i}+\hat{j}|=\sqrt{1^2+1^2}=\sqrt{2}$

$\overrightarrow{a}=\frac{\hat{i}+\hat{j}}{|\hat{i}+\hat{j}|}=\frac{\hat{i}}{\sqrt{2}}+\frac{\hat{j}}{\sqrt{2}}$

$\overrightarrow{b}=\frac{\hat{i}-\hat{j}}{|\hat{i}-\hat{j}|}$

$|\hat{i}-\hat{j}|=\sqrt{1^2+{(-1)}^2}=\sqrt{2}$

$\overrightarrow{b}=\frac{\hat{i}-\hat{j}}{|\hat{i}-\hat{j}|}=\frac{\hat{i}}{\sqrt{2}}-\frac{\hat{j}}{\sqrt{2}}$

Step 2: Find the dot product of $\overrightarrow{A}$ with $\overrightarrow{a}$

$\overrightarrow{A}.\overrightarrow{a}=(2\hat{i}+3\hat{j}).(\frac{\hat{i}}{\sqrt{2}}+\frac{\hat{j}}{\sqrt{2}})$

$=\frac{2}{\sqrt{2}}+\frac{3}{\sqrt{2}}=\frac{5}{\sqrt{2}}$

Step 3: Find the dot product of $\overrightarrow{A}$ with $\overrightarrow{b}$

$\overrightarrow{A}.\overrightarrow{b}=(2\hat{i}+3\hat{j}).(\frac{\hat{i}}{\sqrt{2}}-\frac{\hat{j}}{\sqrt{2}})$

$=\frac{2}{\sqrt{2}}-\frac{3}{\sqrt{2}}=\frac{-1}{\sqrt{2}}$

Hence, the right answer is $\frac{5}{\sqrt{2}},\frac{-1}{\sqrt{2}}$