Vector multiplication

The vector product is written in the form a x b, and is usually called the cross product of two vectors. In this case, we are multiplying the vectors and instead of getting a scalar quantity, we will get a vector quantity. This is the trickiest of the vector computations we'll be dealing with, as it is not commutative and involves the use of the right-hand rule, which I will get to shortly.

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 theta

When 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.

Right hand rule:  Determining the direction of the Vector cross product

If you have a x b, as in the image to the right, you will place your right hand along the length ofb 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 aand 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.

Vector X Vector = Vector

The scalar product of two vectors is a way to multiply them together to obtain a scalar quantity. This is written as a multiplication of the two vectors, with a dot in the middle representing the multiplication. As such, it is often called the dot product of two vectors.

To calculate the dot product of two vectors, you consider the angle between them, as shown in the diagram. In other words, if they shared the same starting point, what would be the angle measurement (theta) between them. The dot product is defined as:

a * b = ab cos theta

Multiply the magnitudes of the two vectors, then multiply by the cosine of the angle of separation. Though a and b - the magnitudes of the two vectors - are always positive, cosine varies so the values can be positive, negative, or zero.

When the vectors are perpendicular (or theta = 90 degrees), cos theta will be zero. Therefore, the dot product of perpendicular vectors is always zero. When the vectors are parallel (or theta= 0 degrees), cos theta is 1, so the scalar product is just the product of the magnitudes.

Vector * Vector = Scalar