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Direction Cosines and Direction Ratios — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDirection Cosines and Direction Ratios3 Marks
Question
If a unit vector $\vec{\text{a}}$ makes an angle $\frac{\pi}{3}$ with $\hat{\text{i}},\frac{\pi}{4}$ with $\hat{\text{j}}$ and an acute angle $\theta$ with $\hat{\text{k}}$, and ,then find the value of $\theta$.
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Answer

Scince a unit vector makes an angle of $\frac{\pi}{3}$ with$\hat{\text{i}}$, $\frac{\pi}{4}$ with $\hat{\text{j}}$ andan acute angle $\theta$ with $\hat{\text{k}},\text{l}=\cos\frac{\pi}{3}$ or $\frac{\pi}{4}$ or $\frac{1}{\sqrt{2}}$and $\text{n}=\cos\theta$.
We know
$\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\Big(\frac{1}{2}\Big)^2+\Big(\frac{1}{\sqrt{2}}\Big)^2+\cos^2\theta=1$
$\Rightarrow\frac{1}{4}+\frac{1}{2}+\cos^2\theta$
$\Rightarrow\cos^2\theta=\frac{1}{4}$
$\Rightarrow\cos^2\theta=\frac{1}{2}$
$\Rightarrow\frac{\pi}{3}$
Thus, the vector $\vec{\text{a}}$ makes an angle of $\frac{\pi}{3}$ with $\hat{\text{k}}$.

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