Question

If $A=3\hat{i}+4\hat{j}$ and $B=7\hat{i}+24\hat{j}$, the vector having the same magnitude as $B$ and parallel to $A$ is

• A. $5\hat{i}+20\hat{j}$
• B. $15\hat{i}+10\hat{j}$
• C. $20\hat{i}+15\hat{j}$
• D. $15\hat{i}+20\hat{j}$

Answer 1

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

$\text{Step 1: Unit vector along}\overrightarrow{A}$

For a vector parallel to $\overrightarrow{A}$, we need to find a unit vector along $\overrightarrow{A}$

Unit vector: $\hat{A}=\frac{\overrightarrow{A}}{|\overrightarrow{A}|}=\frac{3\hat{i}+4\hat{j}}{\sqrt{(3{)}^2+(4{)}^2}}=\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j}$

$\text{Step 2: Required vector}$

Now we need the vector parallel to $\overrightarrow{A}$ to have a magnitude of $\overrightarrow{B}$

$|\overrightarrow{B}|=\sqrt{(7{)}^2+(24{)}^2}=25$

$\Rightarrow$ Required vector $=25\times \hat{A}=25(\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j})$

$=15\hat{i}+20\hat{j}$

Hence, Option(D) is correct.