# Condition for Colplanarity of Four Points

Following are the steps to find out whether points A, B, C and D having poisition vectors $$\vec{A}$$, $$\vec{B}$$, $$\vec{C}$$ and $$\vec{D}$$ are coplanar:

1. Calulate vectors $$\vec{AB}$$, $$\vec{AC}$$ and $$\vec{AD}$$ such that
$$\vec{AB} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$
$$\vec{AC} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$$
$$\vec{AD} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$$
2. The points are coplanar if the following is true
$$\vec{AB}\cdot(\vec{AC} \times \vec{AD})= \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\end{vmatrix}=0$$
Alternatively, the four points (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) and (x4,y4,z4) are coplanar if the following is true
$$\begin{vmatrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \end{vmatrix}=0$$