Question

Find a vector in the direction of vector $2\hat{i}-3\hat{j}+6\hat{k}$ which has magnitude 21 units. [CBSE 2014]

Answer 1

Let $\overrightarrow{a}=2\hat{i}-3\hat{j}+6\hat{k}$.

$\therefore \left|\overrightarrow{a}\right|=\sqrt{2^2+{\left(-3\right)}^2+6^2}=\sqrt{4+9+36}=\sqrt{49}=7$

Unit vector in the direction of $\overrightarrow{a}=\frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}=\frac{2\hat{i}-3\hat{j}+6\hat{k}}{7}$

∴ Vector in the direction of vector $\overrightarrow{a}$ which has magnitude 21 units
$$\begin{gathered} =21\times \left(\frac{2\hat{i}-3\hat{j}+6\hat{k}}{7}\right) \\ =3\left(2\hat{i}-3\hat{j}+6\hat{k}\right) \\ =6\hat{i}-9\hat{j}+18\hat{k} \end{gathered}$$