Multiplying Vectors

1. The Scalar Product (often called dot product)

When we add vectors we always get a new vector, namely  When we multiply vectors we get either a scalar or vector. There are two types of vector multiplication called scalar products or vector product. (Sometimes also called dot product or cross product). The scalar product is defined as

             (1.1)

where a and b are the magnitude of  and  respectively and ø is the angle between and . The whole quantity . = ab cos ø is a scalar, i.e. it has magnitude only. Halliday the scalar product is the product of the magnitude of one vector times the component of the other vector along the first vector. Based on our definition (1.1) we can work out the scalar products of all of the unit vectors.

Thus we have and. Now any vector can be written in terms of unit vectors as  and . Thus the scalar product of any two arbitrary vectors is

Thus we have a new formula for scalar product, namely

   (1.2)

which has been derived from the original definition (1.2)

using unit vectors. What's the good of all this? Well for one thing it's now easy to figure out the angle between vectors, as the next example shows.

2. The Vector Product

In making up the definition of vector product we have to define its magnitude and direction. The symbol for vector product is  Given that the result is a vector let's write The magnitude is defined as

and the direction is defined to follow the right hand rule. ( = thumb,  = forefinger,  = middle finger.)

Thus we have

and

Thus the vector product of any two arbitrary vectors is

which gives a new formula for vector product, namely