Question

If $\overrightarrow{a}=3\hat{i}+4\hat{j}$ and $\overrightarrow{b}=7\hat{i}+24\hat{j}$, then find the vector having the same magnitude as $\overrightarrow{b}$ and same direction as $\overrightarrow{a}$.

• A. $20\hat{i}+15\hat{j}$
• B. $15\hat{i}+18\hat{j}$
• C. $15\hat{i}+20\hat{j}$
• D. $18\hat{i}+15\hat{j}$

Answer 1

The correct option is C $15\hat{i}+20\hat{j}$

Given, $\overrightarrow{a}=3\hat{i}+4\hat{j}$ and $\overrightarrow{b}=7\hat{i}+24\hat{j}$
To get direction we will find unit vector in the direction of $\overrightarrow{a},$
$\hat{a}=\frac{\overrightarrow{a}}{|\overrightarrow{a}|}=\frac{3\hat{i}+4\hat{j}}{\sqrt{3^2+4^2}}=\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j}$
So, required vector $=|\overrightarrow{b}|\hat{a}$
$=\sqrt{7^2+{24}^2}(\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j})$
$=\sqrt{625}(\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j})$
$|\overrightarrow{b}|\hat{a}=15\hat{i}+20\hat{j}$