Question

If \[\overrightarrow{a} = \hat{i} + 2 \hat{j} - 3 \hat{k} \text{ and }\overrightarrow{b} = 2 \hat{i} + 4 \hat{j} + 9 \hat{k} ,\]  find a unit vector parallel to \[\overrightarrow{a} + \overrightarrow{b}\].

Answer 1

Given: \[\overrightarrow{a} = \hat{i} + 2 \hat{j} - 3 \hat{k} , \overrightarrow{b} = 2 \hat{i} + 4 \hat{j} + 9 \hat{k} \]
\[\text{ Now, }\overrightarrow{a} + \overrightarrow{b} = 3 \hat{i} + 6 \hat{j} + 6 \hat{k} \]
\[\left| \overrightarrow{a} + \overrightarrow{b} \right| = \sqrt{3^2 + 6^2 + 6^2} = \sqrt{9 + 36 + 36} = \sqrt{81} = 9\]
Unit vector parallel to \[\overrightarrow{a} + \overrightarrow{b} = \frac{\overrightarrow{a} + \overrightarrow{b}}{\left| \overrightarrow{a} + \overrightarrow{b} \right|} = \frac{3 \hat{i} + 6 \hat{j} + 6 \hat{k}}{9} = \frac{1}{9} \times 3\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right) = \frac{1}{3}\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right)\]