Question

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

Answer 1

Given vector is a $\overrightarrow{a}=\hat{i}-2\hat{j}$

Magnitude of $\overrightarrow{a}=\sqrt{1^2+(-2{)}^2}$

$|\overrightarrow{a}|=\sqrt{1+4}=\sqrt{5}$

Unit vector (i.e., vector having magnitude $1$ ) in the direction of $\overrightarrow{a}$

$\Rightarrow \hat{a}=\frac{\overrightarrow{a}}{|\overrightarrow{a}|}$

$\Rightarrow \hat{a}=\frac{1}{\sqrt{5}}[\hat{i}-2\hat{j}]$
Now, vector having magnitude $7$ in direction of $\hat{a}$

$=7\times \hat{a}$

$=7\times \frac{1}{\sqrt{5}}[\hat{i}-2\hat{j}]$

Thus, required vector is $\frac{7\hat{i}}{\sqrt{5}}-\frac{14\hat{j}}{\sqrt{5}}$