First slide

Vectors

Question

If ${\hat{u}}_{1}$ and ${\hat{u}}_{2}$ are two unit vectors and $| {\hat{u}}_{1} \times {\hat{u}}_{2} | = \sqrt{3} ({\hat{u}}_{1} . {\hat{u}}_{2})$ then the value of ${\hat{u}}_{1} + {\hat{u}}_{2}$ is

Moderate

Solution

$| {\hat{u}}_{1} | | {\hat{u}}_{2} | \sin θ = \sqrt{3} | {\hat{u}}_{1} | | {\hat{u}}_{2} | \cos θ \Rightarrow \tan θ = \sqrt{3} \Rightarrow θ = 60^{0}$
$∴ | {\hat{u}}_{1} + {\hat{u}}_{2} | = {[{| {\hat{u}}_{1} |}^{2} + {| {\hat{u}}_{2} |}^{2} + 2 | {\hat{u}}_{1} | | {\hat{u}}_{2} | . \cos θ]}^{1 / 2}$
$= {(1 + 1 + 2.1.1. \cos 60^{o})}^{1 / 2} = \sqrt{3}$

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