Question

A unit vector $\overrightarrow{a}$makes angles $\frac{\mathrm{\pi }}{4}\mathrm{and}\frac{\mathrm{\pi }}{3}$ with $\hat{i}$ and $\hat{j}$ respectively and an acute angle θ with $\hat{k}$. Find the angle θ and components of $\overrightarrow{a}$.

Answer 1

$$\begin{gathered} \text{Let}\overrightarrow{a}\text{=}{a}_{1}i\hat{}+a_2\hat{j}{\mathrm{+a}}_3\hat{k},\text{where}{a}_1,a_2\text{and}{a}_3\text{are components of}\overrightarrow{a}\text{.} \\ \Rightarrow {a_1}^2+{a_2}^2+{a_3}^2=1\text{(Because}\overrightarrow{a}\text{is a unit vector)}...\left(1\right) \\ \text{Now,} \\ \overrightarrow{a}.\hat{i}=a_1 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\hat{i}\right|\cos \frac{\mathrm{\pi }}{4}=a_1\text{(Because the angle between}\overrightarrow{a}\text{and}\hat{i}\text{is}\frac{\mathrm{\pi }}{4}\text{)} \\ \Rightarrow \left(1\right)\left(1\right)\frac{1}{\sqrt{2}}=a_1\text{(Because}\overrightarrow{a}\text{and}\hat{i}\text{are unit vectors)} \\ \Rightarrow a_1=\frac{1}{\sqrt{2}} \\ \text{Again,} \\ \overrightarrow{a}.\hat{j}=a_2 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\hat{i}\right|\cos \frac{\mathrm{\pi }}{3}=a_2\text{(Because the angle between}\overrightarrow{a}\text{and}\hat{i}\text{is}\frac{\mathrm{\pi }}{3}\text{)} \\ \Rightarrow \left(1\right)\left(1\right)\frac{1}{2}=a_2\text{(Because}\overrightarrow{a}\text{and}\hat{i}\text{are unit vectors)} \\ \Rightarrow a_2=\frac{1}{2} \\ \text{Now from (1),} \\ {\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2+{a_3}^2=1 \\ \Rightarrow \frac{1}{2}+\frac{1}{4}+{a_3}^2=1 \\ \Rightarrow \frac{3}{4}+{a_3}^2=1 \\ \Rightarrow {a_3}^2=\frac{1}{4} \\ \Rightarrow a_3=\frac{1}{2} \\ \text{Now,} \\ \overrightarrow{a}.\hat{k}=a_3 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\hat{k}\right|\cos \mathrm{\theta }=\frac{1}{2}\text{(Because the angle between}\overrightarrow{a}\text{and}\hat{k}\text{is}\mathrm{\theta }\text{)} \\ \Rightarrow \left(1\right)\left(1\right)\cos \mathrm{\theta }=\frac{1}{2}\text{(Because}\overrightarrow{a}\text{and}\hat{i}\text{are unit vectors)} \\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{2}\right)=\frac{\mathrm{\pi }}{3} \\ \mathrm{And} \\ \overrightarrow{a}\text{=}\frac{1}{\sqrt{2}}i\hat{}+\frac{1}{2}\hat{j}+\frac{1}{2}\hat{k} \end{gathered}$$