How to chack that the given vectors are in the same Plane?

Answer 1

A common plane can always be found for two vectors. If you have three however, that is not always the case. If you have three vectors u, v, and w:

1) Take the cross product of u and v. It will yield a vector n that is normal to both of them and any vector in the same plane as they are. Therefore if w is in the same plane as u and v, then n is normal to w as well.

2) Take the dot product of n and w. If they are perpendicular the dot product will be zero and all three vectors are in the same plane. If the dot product is not zero then w is not in the same plane as u and v.

Answer 2

For example;
assuming planes of x,y, and z, cartesian system
vectors of a,b,c

Since there is no vector in the z coordinate, the following vectors are BOTH in the z plane;
3ax+2by
4ax+8by

Since there is no vector in x coordinate, the following vectors are BOTH in the x plane;
3by+2cz
1by+1cz

As far as checking planes that are not x,y, or z plane, but somewhere in between, I'd have to get out my math books and do a little review. You would most likely have to do something similar, in the first case for example I gave the elements a 0 value in the z vector, if they all had a 1 in the z vector they would still be in the same plane (1cz);

3ax+2by+1cz
4ax+8by+1cz

Answer 3

http://en.wikipedia.org/wiki/Coplanarity