Vector Multiplication

(Click here for a discussion about vector addition.)
(Click here for examples of resolving 3-D vectors into components.)

There are two vector operations involving multiplication:

1. Scalar product is given by the equation

A·B = AB cos q
= AxBx + AyBy + AzBz.

2. Vector product is given by the equations

AxB = C and |C| = AB sin q.

The meaning of the scalar product operation is: take the magnitude of the first vector, and multiply by the component of the second vector along the direction of the first. The result is just a number, in the same units as (A)(B). There are two uses for this operation in physics:

  1. expressing degree of "parallel-ness" between one vector and another. The larger the scalar product, the closer the vectors are to being along the same direction.
  2. calculating the angle q between two vectors that are known only in unit-vector notation. The diagram above shows that there is no simple geometric formula that would allow you to calculate the angle q because it is not in the xy, yz or xz planes.

The meaning of the vector product operation is: take the magnitude of the first vector, and multiply by the component of the second vector perpendicular to the direction of the first and express the result as a new vector C pointing perpendicular to the plane of A and B. To determine which perpendicular direction to point C requires the right-hand rule: put the fingers of your right hand in the direction of A, then rotate your wrist in such a way that you can curl those fingers from the direction of A into the direction of B; your thumb will then point in the direction of C. In the diagram here, the vector C points to the left and away from you, and it is perpendicular to the plane formed by A and B.

The vector C in unit-vector notation is:

C = (AyBz-AzBy)i + (AzBx-AxBz)j + (AxBy-AyBx)k.

In the diagram, the vector C has negative x and y components; this is built into the equation above in that there is a difference between products of coefficients that determines the component's sign.

It is even more difficult to visualize the vector product (a.k.a. cross-product) in terms of any changes you make to A or B. Click here for a Java-based demonstration of changing vector products. There are two uses for the vector product operation in physics:

  1. expressing degree of perpendicularity between two vectors; the larger the magnitude of C, the more perpendicular A and B are.
  2. using the third vector's direction as an axis around which something rotates (see the discussion of rotational motion and torque).

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