Hi, I'm stuck on the following question:
find vectors parallet to the xy plane that are perpendicular to the vector (-2,3,5)
I realise that the z co-ord in these 2 vectors would be the same so I let m and n be the 2 vectors, where m=(a,b,c) and n=(d,e,c)
I then tried to find the cross product of m and n as that would equate to (-2,3,5) but came out with 5 unknowns and 3 equations...
Project onto the xy plane by setting the z coordinate to 0, and then rotate through a right angle. So you get first (-2,3,0) and then (-3,-2,0) or (3,2,0) depending which way round you went. Either vector is a non-zero vector that has a zero dot product with -2,3,5, so it must be perpendicular to it, and both lie in the xy plane. There is no way a vector with a non-zero z part is going to be parallel to the xy plane. Of course you might like to think of the specific instance of the vector that is the directed line segment from (-2,3,5) to (-5,1,5).