A unit vector a makes angles 4 and 3 with i and j respectively an… - Vidyadip

Question

A unit vector $\vec{\text{a}}$ makes angles $\frac{\pi}{4}$ and $\frac{\pi}{3}$ with $\hat{\text{i}}$ and $\hat{\text{j}}$ respectively and an acute angle $\theta$ with $\hat{\text{k}}$. find the angle $\theta$ and components of $\vec{\text{a}}$ .

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Answer

Let $\vec{\text{a}}=\text{a}_{1}\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},$ where $\text{a}_1,\text{a}_2$ and $\text{a}_3$ are components of $\vec{\text{a}.}$
$\Rightarrow{\text{a}_1}^2+{\text{a}_2}^2+{\text{a}_3}^2=1$ (Because $\vec{\text{a}}$ is a unit vector) ...(1)
Now,
$\vec{\text{a}}.\hat{\text{i}}=\text{a}_1$
$\Rightarrow|\vec{\text{a}}||\hat{\text{i}}|\cos\frac{\pi}{4}=\text{a}_1$ (Because the angle between $\vec{\text{a}}$ and $\hat{\text{i}}$ is $\frac{\pi}{4}$)
$\Rightarrow(1)(1)\frac{1}{\sqrt{2}}=\text{a}_1$ (Because $\vec{\text{a}}$ and $\hat{\text{i}}$ are unit vectors)
$\Rightarrow \text{a}_1=\frac{1}{\sqrt{2}}$
Again,
$\vec{\text{a}}.\hat{\text{j}}=\text{a}_2$
$\Rightarrow|\vec{\text{a}}||\hat{\text{i}}|\cos\frac{\pi}{3}=\text{a}_2$ (Becasue the angle between $\vec{\text{a}}$ and $\hat{\text{i}}$ is $\frac{\pi}{3}$)
$\Rightarrow(1)(1)\frac{1}{2}=\text{a}_2$ (Because $\vec{\text{a}}$ and $\hat{\text{i}}$ are unite vectors)
$\Rightarrow{\text{a}}_2=\frac{1}{2}$
Now from (1),
$\Big(\frac{1}{\sqrt{2}}\Big)^2+\big(\frac{1}{2}\big)^2+{\text{a}_3}^2=1$

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