Question

Four points are always coplanar if

Answer 1

For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane. Two lines in three-dimensional space are coplanar if there is a plane that includes them both.

Answer 2

Answer:

Four points are always  coplanar if triple scalar  product is zero.

Step-by-step explanation:

We have four points and we find the condition for four points are coplanar.

Let A,B,C and D are four points .We have to find Vector AB, AC and AD.

Then we find triple scalar product of  vectors AB,AC and AD.

Triple scalar product=$vec{AB}.(\vec{AC\timesAD})$

To find triple scalar product we use determinant

Suppose A(1,0,-1),B(0,2,3),C(-2,1,1) and D(4,2,3) are four points .

We shall prove that A,B,C and D are coplanar.

We find vector AB,AC and AD

$\vec{AB}$=Coordinate of B -coordinate of A=$-1\hat{i}+2\hat{j}+4\hat{k}$

$\vec{AC}$=Coordinate of C-coordinate of A=$-3\hat{i}+\hat{j}+2\hat{k}$

$\vec{AD}$=Coordinate of D- coordinate of A=$3\hat{i}+2\hat{j}+4\hat{k}$

Now , we find triple scalar product

$\vec{AB}.(\vec{AC}\times\vec{AD})=\begin{vmatrix}-1&2&4\\-3&1&2\\3&2&4\end{vmatrix}$

Expand alon $R_1$

$\vec{AB}.(\vec{AC}\times\vec{AD})$=-1(4-4)-2(-12-6)+4(-6-3)

$\vec{AB}.(\vec{AC}\times\vec{AD})$=0+36-36=0

Hence, triple scalar product is zero therefore, four points A,B,C and D are coplanar.

Four points are always coplanar if triple scalar product is zero.