Question

Find a vector of magnitude of 5 units parallel to the resultant of the vectors \[\vec{a} = 2 \hat{i} + 3 \hat{j} - \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} +\widehat{k} .\]

Answer 1

Given the position vectors \[\vec{a} = 2 \hat{i} + 3 \hat{j} - \hat{k}\] and \[\vec{b} = \hat{i} - 2 \hat{j} + \hat{k}\]
∴ Resultant Vector = \[\vec{a} + \vec{b} = 2 \hat{i} + 3 \hat{j} - \hat{k} + \hat{i} - 2 \hat{j} + \hat{k} = 3 \hat{i} + \hat{j}\] 
So, a unit vector parallel to the resultant vector is \[\frac{3 \hat{i} + \hat{j}}{\left| 3 \hat{i} + \hat{j} \right|} = \frac{3 \hat{i} + \hat{j}}{\sqrt{3^2 + 1^2}} = \frac{3 \hat{i} + \hat{j}}{\sqrt{10}}\]
Hence, required vector = \[5 \times \frac{\left( 3 \hat{i} + \hat{j} \right)}{\sqrt{10}} = \sqrt{\frac{5}{2}}\left( 3 \hat{i} + \hat{j} \right)\]