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lara [203]
3 years ago
6

Four points are always coplanar if

Mathematics
2 answers:
Kaylis [27]3 years ago
8 0

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.

My name is Ann [436]3 years ago
7 0
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.
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