Calculating the Magnitude
Again, we consider two vectors drawn from the same point, with the angle theta between them (see picture to right). We always take the smallest angle, so theta will always be in a range from 0 to 180 and the result will, therefore, never be negative. The magnitude of the resulting vector is determined as follows:
If c = a x b, then c = ab sin thetaWhen the vectors are parallel, sin theta will be 0, so the vector product of parallel (or antiparallel) vectors is always zero. Specifically, crossing a vector with itself will always yield a vector product of zero.
Direction of the Vector
Now that we have the magnitude of the vector product, we must determine what direction the resulting vector will point. If you have two vectors, there is always a plane (a flat, two-dimensional surface) which they rest in. No matter how they are oriented, there's always one plane that includes them both. (This is a basic law of Euclidean geometry.)The vector product will be perpendicular to the plane created from those two vectors. If you picture the plane as being flat on a table, the question becomes will the resulting vector go up (our "out" of the table, from our perspective) or down (or "into" the table, from our perspective)?
The Dreaded Right-Hand Rule
In order to figure this out, you must apply what is called the right-hand rule. When I studied physics in school, I detested the right-hand rule. Flat out hated it. Every time I used it, I had to pull out the book to look up how it worked. Hopefully my description will be a bit more intuitive than the one I was introduced to which, as I read it now, still reads horribly.
If you have a x b, as in the image to the right, you will place your right hand along the length of b so that your fingers (except the thumb) can curve to point along a. In other words, you are sort of trying to make the angle theta between the palm and four fingers of your right hand. The thumb, in this case, will be sticking straight up (or out of the screen, if you try to do it up to the computer). Your knuckles will be roughly lined up with the starting point of the two vectors. Precision isn't essential, but I want you to get the idea since I don't have a picture of this to provide.
If, however, you are considering b x a, you will do the opposite. You will put your right hand along a and point your fingers along b. If trying to do this on the computer screen, you will find it impossible, so use your imagination. You will find that, in this case, your imaginative thumb is pointing into the computer screen. That is the direction of the resulting vector.
The right-hand rule shows the following relationship:
a x b = -b x aNow that you have the means of finding the direction of c = a x b, you can also figure out the components of c:
cx = ay bz - az byNotice that in the case when a and b are entirely in the x-y plane (which is the easiest way to work with them), their z-components will be 0. Therefore, cx & cy will equal zero. The only component of c will be in the z-direction - out of or into the x-y plane - which is exactly what the right-hand rule showed us!
cy = az bx - ax bz
cz = ax by - ay bx
Final Words
Don't be intimidated by vectors. When you're first introduced to them, it can seem like they're overwhelming, but some effort and attention to detail will result in quickly mastering the concepts involved.At higher levels, vectors can get extremely complex to work with. Entire courses in college, such as linear algebra, devote a great deal of time to matrices (which I kindly avoided in this introduction), vectors, and vector spaces. That level of detail is beyond the scope of this article, but this should provide the foundations necessary for most of the vector manipulation that is performed in the physics classroom. If you are intending to study physics in greater depth, you will be introduced to the more complex vector concepts as you proceed through your education.
