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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA1 Mark
Question
If a unit vector $\overrightarrow{\text{a}}$makes angles $\frac{\pi}{3}$with $\hat{\text{i}},\frac{\pi}{4}$with $\hat{\text{j}}$and an acute angle$\theta$ with $\hat{\text{k}},$ then find the value of $\theta.$
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Answer

Let l, m, n, be Direction cosines of $\overrightarrow{\text{a}}$
$\therefore\text{l} =\cos\frac{\pi}{3} =\frac{1}{2};\text{m} = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}};\text{n} = \cos\theta$
$\because\text{l}^{2} + \text{m}^{2} + \text{n}^{2} = 1 $
$\Rightarrow\bigg(\frac{1}{2}\bigg)^{2} + \bigg(\frac{1}{\sqrt{2}}\bigg)^{2} + \cos^{2}\theta = 1 $
$\Rightarrow\frac{1}{4} + \frac{1}{2} + \cos^{2}\theta = 1 $
$\Rightarrow\cos^{2}\theta = 1 - \bigg(\frac{1}{4} + \frac{1}{2}\bigg) = 1 - \frac{3}{4} = \frac{1}{4}$
$\Rightarrow\cos\theta =\frac{1}{2}\Rightarrow\theta = \frac{\pi}{3}.$

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