Since $\hat{i}, \hat{j}$ and $\hat{k}$ are the unit vectors in the direction of $\mathrm{X}, \mathrm{Y}$ and $\mathrm{Z}$ in Cartesian co-ordinate system.

$(i)$ $\hat{i} \cdot \hat{i}=(1)(1) \cos 0^{\circ} \quad[\because|\hat{i}|=1$, and $\hat{i} \| \hat{i}]$

$\therefore \hat{i} \cdot \hat{i}=1 \quad\left[\because \cos 0^{\circ}=1\right]$

similarly $\hat{j} \cdot \hat{j}=1$ and $\hat{k} \cdot \hat{k}=1$

$(ii)$ $\hat{i} \cdot \hat{j}=(1)(1) \cos 90^{\circ} \quad[\because|\hat{i}|=1,|\hat{j}|=1$ and $\hat{i} \perp \hat{j}]$

$\therefore \hat{i} \cdot \hat{j}=\hat{j} \cdot \hat{i}=0 \quad\left[\because \cos 90^{\circ}=0\right]$

similarly, $\quad \hat{j} \cdot \hat{k}=\hat{k} \cdot \hat{j}=0$

$\hat{k} \cdot \hat{i}=\hat{i} \cdot \hat{k}=0$