Coplanar vectors with four points.

My teacher taught the class how to do problems dealing with coplanar vectors with three points and now the book is asking for the reader to solve with four points. In the book there is no explanation on how to solve it. The question is found below:

5. Determine if the points are coplanar.
a) $\displaystyle A(-1, 2, 1), B(3, -1, 2), C(1, 4, -3), D(7, 2, 1)$
b) $\displaystyle J(5, 7, -2), K(8, 3, 0), L(4, 10, 1), M(9, 0, -3)$
c) $\displaystyle P(-3, 5, 4), Q(2, 3, 1), R(8, 4, 0), S(3, -1, 2)$

The book says a and c are not coplanar and b is coplanar. How does one solve if it is coplanar?

Given three non-zero vectors $\displaystyle U,~V,~\&~W$ they are co-planar if $\displaystyle W\cdot(U\times V)=0$.

There are four vectors not three. I tried that and it did not work.

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Originally Posted by Barthayn

There are four vectors not three. I tried that and it did not work.

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That is total nonsense.
Any four points determine at most three non-collinear.
In your post you gave points not vectors.
What do you mean? Are the points or vectors?

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Originally Posted by Plato

That is total nonsense.
Any four points determine at most three non-collinear.
In your post you gave points not vectors.
What do you mean? Are the points or vectors?

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With the points given we have to find which are coplanar. There are four points on the graph, which can be made into vectors by taking the magnitude of the points. However, I do not know how to get the answer. The teacher only taught us how to do it with three points.

Plato's post is correct (as usual). The vectors you're seeking to make are not the correct ones. Instead, try constructing three vectors of the form A-B, A-C, A-D, and seeing if those vectors are coplanar.

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Originally Posted by Ackbeet

Plato's post is correct (as usual). The vectors you're seeking to make are not the correct ones. Instead, try constructing three vectors of the form A-B, A-C, A-D, and seeing if those vectors are coplanar.

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thanks for the helpful hint. It didn't cross my mind to try that.

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Originally Posted by Barthayn

thanks for the helpful hint. It didn't cross my mind to try that.

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Pun unintended, I assume? (Wink)