Take the determinant. If the determinant is 0, the vectors are linearly dependent, i.e. coplanar.
Therefore,
Problem:
Verify that the vectors <2,3,1>, <1,-1,0>, and <7,3,2> are coplanar.
Question:
Of course, the easiest way to do this is using the scalar triple product. At the time of my test, however, I was unaware of this. I attempted to find the angle between the vectors, and I got the result that the angle did not exist for any vectors I tried. So, if there is no angle between any of the vectors, does it follow that they are coplanar?
The angle always exists between two nonzero vectors.
v dot w = |v||w|cos(angle)
How can the angle be undefined? (Unless one of |v| or |w| is 0, but that is not the case for this problem)
Also, the angle won't help with being coplanar.
Think about 3 vectors in the xy-plane. They have to be coplanar, but there is no relation between their angles.
Angle between what two vectors? The angle between two non-zero vectors always exists!
You could, as dwsmith did at the end of his post, using the result from the post, show that one of the vectors is a linear combination of the other two. That is, <2,3,1>, <1,-1,0>, and <7,3,2> are co-planar if and only if there exist numbers a and b such that <7, 3, 2>= a<2, 3, 1>+ b<1, -1, 0>. That is the same as saying <7, 3, 2>= <2a+ b, 3a- b, a> or the system of equations 2a+ b= 7, 3a- b= 3, a= 2. Since the last equation tells us immediately that a= 2, the first equation becomes 4+ b= 7 or b= 3. Putting a= 2 and b= 3 in the second equation we get 3(2)- 3= 6- 3= 3 so they also satisfy the second equation. That tells us that the three vectors are co-planar.
Call the three vectors
Use the scalar product to find the angle between each pair of vectors.
The angle between and is:
Find and similarly.
If the vectors are co-planar, then one of the two following things will be true.
1. The sum of two of these angles is equal to the third.For the three given angles, #1 works.
2. The sum of all three angles will be 360° (or radians).
You could do a similar process using the vector product. Then
In this case, criterion #2 would change to: The sum of the three angles being 180°, rather than 360°, would indicate that the vectors are co-planar.