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
\(\begin{vmatrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \end{vmatrix}=0\)