Question

A unit vector in the direction of resultant vector of $\vec{A} = - 2 \hat{i} + 3 \hat{j} + \hat{k} and \vec{B} = \hat{i} + 2 \hat{j} - 4 \hat{k} is$

Moderate

Solution

Here, $\vec{A} = - 2 \hat{i} + 3 \hat{j} + \hat{k}$
$\vec{B} = \hat{i} + 2 \hat{j} - 4 \hat{k}$
The resultant vector of $\vec{A} and \vec{B}$ is
$\vec{R} = \vec{A} + \vec{B} = (2 \hat{i} + 3 \hat{j} + \hat{k}) + (\hat{i} + 2 \hat{j} - 4 \hat{k})$
$= - \hat{i} + 5 \hat{j} - 3 \hat{k}$
$| \vec{R} | = \sqrt{{(- 1)}^{2} + {(5)}^{2} + {(- 3)}^{2}} = \sqrt{1 + 25 + 9} = \sqrt{35}$
Unit vector in the direction of resultant vector of $\vec{A} and \vec{B}$ is
$\hat{R} = \frac{\vec{R}}{| \vec{R} |} = \frac{- \hat{i} + 5 \hat{j} - 3 \hat{k}}{\sqrt{35}}$

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