Question

Write a unit vector in the direction of the sum of the vectors \[\overrightarrow{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}\] and \[\overrightarrow{b} = 2 \hat{i} + \hat{j} - 7 \hat{k}\].

Answer 1

We have,
\[\overrightarrow{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}\] and \[\overrightarrow{b} = 2 \hat{i} + \hat{j} - 7 \hat{k}\].
\[\therefore \overrightarrow{a} + \overrightarrow{b} = \left( 2 \hat{i} + 2 \hat{j} - 5 \hat{k} \right) + \left( 2 \hat{i} + \hat{j} - 7 \hat{k} \right) = 4 \hat{i} + 3 \hat{j} - 12 \hat{k}\]
\[\Rightarrow \left| \overrightarrow{a} + \overrightarrow{b} \right| = \sqrt{4^2 + 3^2 + \left( - 12 \right)^2} = \sqrt{16 + 9 + 144} = \sqrt{169} = 13\]
∴ Required unit vector = \[\frac{\overrightarrow{a} + \overrightarrow{b}}{\left| \overrightarrow{a} + \overrightarrow{b} \right|} = \frac{4 \hat{i} + 3 \hat{j} - 12 \hat{k}}{13} = \frac{4}{13} \hat{i} + \frac{3}{13} \hat{j} - \frac{12}{13} \hat{k}\]