Question: Find a condition for four points in space to lie on a plane. Your answer should be in the form of a determinant which must be equal to zero. Hint: The equation of a plane is of the form$\mathbf{\text{ax}}\mathbf{+}\mathbf{\text{by}}\mathbf{+}\mathbf{\text{cz}}\mathbf{=}\mathbf{\text{d}}$, where a,b,c,dare constants. The four points$\left({\text{x}}_{\text{1}}{\text{,y}}_{\text{1}}{\text{,z}}_{\text{1}}\right)\mathbf{\text{,}}\left({\text{x}}_{\text{2}}{\text{,y}}_{\text{2}}{\text{,z}}_{\text{2}}\right)$etc., are all satisfy this equation. When can you find a,b,c,dnot all zero?

A plane is defined as a flat, two-dimensional surface that extends indefinitely in mathematics. Also,the two-dimensional equivalence of a point, a line, and three-dimensional space is considereda plane.

The determinant is a scalar variable in mathematics that is a function of the entries of a square matrix. It identifies some of the matrix's attributes as well as the linear map that the matrix represents.

Consider four points are$\left({\text{x}}_{\text{1}}{\text{,y}}_{\text{1}}{\text{,z}}_{\text{1}}\right)\text{,}\left({\text{x}}_{\text{2}}{\text{,y}}_{\text{2}}{\text{,z}}_{\text{2}}\right)\text{,}\left({\text{x}}_{\text{3}}{\text{,y}}_{\text{3}}{\text{,z}}_{\text{3}}\right)\text{,}\left({\text{x}}_{\text{4}}{\text{,y}}_{\text{4}}{\text{,z}}_{\text{4}}\right)$

Characterize the four vectors.

$$\begin{gathered} a{x}_1+b{y}_1+{\text{cz}}_1=d \\ a{x}_2+b{y}_2+c{z}_2=d \\ a{x}_3+b{y}_3+c{z}_3=d \\ a{x}_4+b{y}_4+c{z}_4=d \end{gathered}$$

The determinant of the coefficient will be zero when a system of homogenous equations in unknowns will have solutions other than the trivial solution.

Thus, a condition is that the determinant of their coordinate with 1 has to be zero.

$\left|\begin{array}{cccc}{\text{x}}_{\text{1}}& {\text{y}}_{\text{1}}& {\text{z}}_{\text{1}}& 1\\ {\text{x}}_2& {\text{y}}_2& {\text{z}}_2& 1\\ {\text{x}}_3& {\text{y}}_3& {\text{z}}_3& 1\\ {\text{x}}_4& {\text{y}}_4& {\text{z}}_4& 1\end{array}\right|=0$$\left|\begin{array}{cccc}{\text{x}}_{\text{1}}& {\text{y}}_{\text{1}}& {\text{z}}_{\text{1}}& 1\\ {\text{x}}_2& {\text{y}}_2& {\text{z}}_2& 1\\ {\text{x}}_3& {\text{y}}_3& {\text{z}}_3& 1\\ {\text{x}}_4& {\text{y}}_4& {\text{z}}_4& 1\end{array}\right|=0$