Question

Find a unit vector in the direction of the resultant of the vectors
\[\hat{i} - \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} - 2 \hat{k} \text{ and }\hat{i} + 2 \hat{j} - 2 \hat{k} .\]

Answer 1

Given: \[\vec{a} =\hat{i} - \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} - 2 \hat{k} \text{ and } \vec{c} = \hat{i} + 2 \hat{j} - 2 \hat{k} .\] are the position vectors.
Then, Resultant of the vectors = \[\vec{a} + \vec{b} + \vec{c}\]
\[= \hat{i} - \hat{j} + 3 \hat{k} + 2 \hat{i} + \hat{j} - 2 \hat{k} + \hat{i} + 2 \hat{j} - 2 \hat{k} \]
\[ = 4 \hat{i} + 2 \hat{j} - \hat{k}\]
So,
\[\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{4^2 + 2^2 + 1^2} = \sqrt{16 + 4 + 1} = \sqrt{21}\]|
∴ Unit vector in the direction of the resultant vector =

\[\frac{\vec{a} + \vec{b} + \vec{c}}{\left| \vec{a} + \vec{b} + \vec{c} \right|} = \frac{1}{\sqrt{21}}\left( 4 \hat{i} + 2 \hat{j} - \hat{k} \right)\]