If a vector has direction angles 45° and 60°, find the third dire… - Vidyadip

Question

If a vector has direction angles 45° and 60°, find the third direction angle.

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Answer

Let $\alpha, \beta, \gamma$ be the angles made by the vector with positive directions of $X, Y, Z$ axes respectively.
$\therefore \alpha=45^{\circ}, \beta=60^{\circ}$
We know that,
$ \because \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1$
$\therefore \cos ^2 45^{\circ}+\cos ^2 60^{\circ}+\cos ^2 r=1 $
$\therefore\left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{2}\right)^2+\cos ^2 \gamma=1$
$\therefore \frac{1}{2}+\frac{1}{4}+\cos ^2 \gamma=1$
$ \therefore \cos ^2 \gamma=1-\frac{1}{2}-\frac{1}{4}$
$\therefore \cos ^2 \gamma=\frac{1}{4} $
$\therefore \cos ^2 \gamma=\frac{1}{4}$
$\therefore \cos \gamma= \pm \frac{1}{2}$
$\therefore \cos \gamma=\frac{1}{2}$ or $\cos \gamma=-\frac{1}{2}$
$\therefore \cos \gamma=\frac{\pi}{3}$ or $\cos \gamma=-\frac{\pi}{3}$
$\therefore \cos \left(\pi-\frac{\pi}{3}\right)=\cos \frac{2 \pi}{3}$
$\therefore \gamma=\frac{\pi}{3}$ or $\gamma=\frac{2 \pi}{3}$
Hence, the third direction angle is $\frac{\pi}{3}$ or $\frac{2 \pi}{3}$

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