I want to know how calculate the angle between two vectors and both are not in same plane, which means that they don't intersect at any point?
For example how do I calculate angle between AB and EF in this figure
https://docs.google.com/file/d/0B6ehLXK0eP7sV3ByekxQZlZnQ1U/edit
I also have a doubt about the dot product. Can it be only used when the two vectors lie in the same plane ??
Thank you
MN
Best Answer
You are confusing rays with vectors. Vectors represent length and direction. They are, thus, relative quantities. To say that two vectors aren't in the same plane is meaningless, because they have no origin. You can, however, say that two non-zero vectors always describe a plane.
Rays, on the other hand, have magnitude, direction, and origin, and certainly, the angle between them is well defined. So, your question seems to be more about whether the dot product can be used to find the angle between two rays, and the short answer is, yes.
While you are correct that two rays might not lie on intersecting lines, you can always find two parallel rays that do lie on intersecting lines. To see this, just draw a line segment from the origin of one ray, which we'll call ray A, to the origin of the other, which we'll call ray B. Then, draw another line segment of equal length, starting from the "tip" of ray A parallel to the first line segment. Finally, join the free ends of these two line segments, and you will get a ray parallel to ray A that shares its origin with ray B, as in the following crudely drawn picture:
Now, you have two co-planar rays, and you can use dot product confidently to your heart's content. Just make sure that you use the vectors of the rays, and not their origins.