Question

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

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
\[= 21 \times \left( \frac{2\hat{i} - 3 \hat{j} + 6 \hat{k}}{7} \right)\]
\[ = 3\left( 2 \hat{i} - 3 \hat{j} + 6 \hat{k} \right)\]
\[ = 6 \hat{i} - 9 \hat{j} + 18 \hat{k}\]