Question

If $\overrightarrow{a}=\hat{i}-\hat{j}$ and$\overrightarrow{b}=3\hat{i}+4\hat{j}$, the vector having the same magnitude as $\overrightarrow{b}$ and parallel to $\overrightarrow{a}$ is

• A. $\frac{5}{\sqrt{2}}(\hat{i}-\hat{j})$
• B. $\frac{5}{\sqrt{2}}(\hat{i}+\hat{j})$
• C. $5(\hat{i}-\hat{j})$
• D. $5(\hat{i}+\hat{j})$

Answer 1

The correct option is A $\frac{5}{\sqrt{2}}(\hat{i}-\hat{j})$

Magnitude of $\overrightarrow{b}$ is $\sqrt{3^2+4^2}=5$
Unit vector in direction of $\overrightarrow{a}$ is $\frac{\overrightarrow{a}}{|\overrightarrow{a}|}$
$\hat{a}=\frac{\hat{i}-\hat{j}}{\sqrt{1^2+1^2}}=\frac{1}{\sqrt{2}}(\hat{i}-\hat{j})$
Now, vector having magnitude same as $\overrightarrow{b}$ and parallel to $\overrightarrow{a}$ is
$=|b|\times \hat{a}$
$=5\times \frac{1}{\sqrt{2}}(\hat{i}-\hat{j})$
$=\frac{5}{\sqrt{2}}(\hat{i}-\hat{j})$