HELP!!! determine if the following points are coplanar????

Question

A(3,1,0)
B(2,-3,1)
C(-1,0,4)
D(5,-6,-2)


i THINK your supposed to make 3 vectors then do the system equation elimniation thing and see if Left Side = Right Side...

except when i do it, its wrong!


can someone show the work??

Answer 1

There are several way to do the problem.

Let's start with vector algebra:

the vectors AB and AC are independent, and in the plane defined by the points A, B, and C.

The cross product, X = AB x AC is a vector perpendicular to both and hence perpendicular to the plane.

If the point D is in the same plane as A, B, and C, then the vector AD will be a linear combination of AB and AC, so the dot product of AD and the cross-product vector X will be 0. Since AD is not the zero vector, D is coplanar if and only if this dot product is 0

A web search found this worked out example (with different numbers):
http://jtaylor1142001.net/calcjat/Solutions/VCrossProduct/VCP4PtsCoplanar.htm

You can also use matrices and computing their determinants, but it seems you are less than happy with that approach. Still, here is a link:

http://mathworld.wolfram.com/Coplanar.html

(In the end, it is the same computation.)

Answer 2

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