Triple Scalar Product: using it to determine if FOUR points are co-planar

[HollyAnne]

Determine whether the points P₁ = (-1, 1, 2), P₂ = (3, 3, 4), P₃ = (2, -2, 10), and P₄ = (0, 2, 2) lie on the same plane using the scalar triple product

I know how to do this with 3 points a (bxc), but am not sure how to implement with 4 points?

 

[pka]

Determine whether the points P₁ = (-1, 1, 2), P₂ = (3, 3, 4), P₃ = (2, -2, 10), and P₄ = (0, 2, 2) lie on the same plane using the scalar triple product

Let:
\(\displaystyle \begin{array}{l}\vec{Q} = {P_1} - {P_2} = \left\langle {4,2,2} \right\rangle \\\vec{R }= {P_1} - {P_3} = \left\langle {3, - 3,8} \right\rangle \\\vec{S} = {P_1} - {P_4} = \left\langle {1,1,0} \right\rangle \end{array}\)

\(\displaystyle \vec{Q}\times(\vec{R}\times\vec{S})=(\vec{Q}\cdot \vec{S} )\vec{R}-(\vec{Q}\cdot\vec{R})\vec{S}=~?\)

OR IS IT
\(\displaystyle \vec{Q}\cdot(\vec{R}\times\vec{S})=~?\)