Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$ such that $|\vec{a}|=1$ $\Rightarrow \quad\left(a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\right) \cdot \hat{i}=|\vec{a}||\hat{i}| \cos \frac{\pi}{3}$
$$
\Rightarrow \quad\left(a_1\right)(1)=(1)(1) \frac{1}{2} \Rightarrow a_1=\frac{1}{2}
$$
Also $\left(a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\right) \hat{j}=|\vec{a}||\hat{j}| \cos \frac{\pi}{4}$
$$
\Rightarrow \mathrm{a}_2=\frac{1}{\sqrt{2}} \text { As }|\vec{a}|=1 \Rightarrow \mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3^2=1
$$
$$
\Rightarrow\left(\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2+\mathrm{a}_3^2=1
$$
$$
\Rightarrow \quad \mathrm{a}_3=\frac{1}{2} \quad \therefore \quad \overrightarrow{\mathrm{a}}=\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{2} \hat{\mathrm{k}}
$$
and $\left(\frac{1}{2} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\frac{1}{2} \hat{k}\right) \cdot \hat{k}=\left|\frac{1}{4}+\frac{1}{2}+\frac{1}{4}\right||1| \cos \theta$
$$
\Rightarrow \quad \cos \theta=\frac{1}{2} \text { i.e., } \theta=\frac{\pi}{3}
$$
$\therefore \quad$ components of $\vec{a}$ are $\frac{1}{2} \hat{i}, \frac{1}{\sqrt{2}} \hat{j}, \frac{1}{2} \hat{k}$