Obtain the scalar product of unit vectors in the Cartesian coordinate system.

(N/A) In the Cartesian coordinate system,$\hat{i}, \hat{j}$,and $\hat{k}$ are unit vectors along the $X, Y$,and $Z$ axes,respectively.
$(i)$ For parallel unit vectors:
$\hat{i} \cdot \hat{i} = |\hat{i}| |\hat{i}| \cos 0^{\circ} = (1)(1)(1) = 1$
Similarly,$\hat{j} \cdot \hat{j} = 1$ and $\hat{k} \cdot \hat{k} = 1$.
$(ii)$ For orthogonal unit vectors:
$\hat{i} \cdot \hat{j} = |\hat{i}| |\hat{j}| \cos 90^{\circ} = (1)(1)(0) = 0$
Similarly,$\hat{j} \cdot \hat{k} = 0$ and $\hat{k} \cdot \hat{i} = 0$.
Thus,the scalar products are:
$\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$
$\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$