Area of a Triangle, If three vertices are given taken in anticlockwise direction. Three vertices are given. We can find the area using the determinant. Can someone explain it to me why the number will be a positive number, if vertices are chosen in anticlockwise direction.

Using rschwieb's diagram, change the coordinate system so that $(x_1,y_1)$ is at the origin, and so that $x_2>0$ and $y_2 = 0$. (This obviously doesn't change the area). Then the three points are $(0,0)$, $(x_2,0)$, and $(x_3,y_3)$. Since the angle between $\mathbf{a}$ and $\mathbf{b}$ is strictly between $0$ and $\pi$, we know that $y_3 > 0$. Then
$$\begin{vmatrix} x_2 & 0 \\ x_3 & y_3\end{vmatrix} = x_2y_3 > 0.$$
