Find a vector in the direction of the $\overrightarrow{a}$ = $\hat{i} - 2 \hat{j}$ that has magnitude 7 units.

$7 \hat{i} - 14 \hat{j}$

$\dfrac{7}{\sqrt5} \hat{i} - \dfrac{14}{\sqrt5} \hat{j}$

$\dfrac{7}{\sqrt5} \hat{i} + \dfrac{14}{\sqrt5} \hat{j}$

$7 \hat{i} + 14 \hat{j}$

Karnataka PU MQP 2023

Solution

The correct answer is

$\dfrac{7}{\sqrt5} \hat{i} - \dfrac{14}{\sqrt5} \hat{j}$

Explanation

Given vector

$\overrightarrow{a}$ =

$\hat{i} - 2 \hat{j}$

Unit vector in the direction of

$\hat{a}$ =

$\dfrac{\overrightarrow{a}}{|\overrightarrow{a}|}$ =

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

$\dfrac{\hat{i} - 2 \hat{j}}{\sqrt5}$

∴ Vector in the direction

$\overrightarrow{a}$ that has magnitude 7 units =

$7 \left( \dfrac{\hat{i} - 2 \hat{j}}{\sqrt5} \right)$ =

$\dfrac{7}{\sqrt5} \hat{i} - \dfrac{14}{\sqrt5} \hat{j}$

Find a vector in the direction of the →a\overrightarrow{a} = ˆi−2ˆj\hat{i} - 2 \hat{j} that has magnitude 7 units.