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# 4.3 Properties of the scalar product

Activity 20 illustrates two important properties of the scalar product. First, if two non-zero vectors are perpendicular, then their scalar product is zero. This is because if and are perpendicular, then

This also works the other way, so that if the scalar product of two non-zero vectors is zero, then the vectors are perpendicular. This is because if and are non-zero vectors, then the only way that can be equal to zero is if . This implies that .

The second property is that the scalar product of a vector with itself is equal to the square of the magnitude of the vector. This is because if is any non-zero vector, then the angle between and itself is 0°, so

These, and other, properties of the scalar product in the following list can all be proved using the definition of the scalar product in a similar way.

## Scalar product properties

The following properties hold for all vectors , and , and every scalar .

1. If and are non-zero and perpendicular, then and .

We can use these properties to simplify expressions containing scalar products of vectors.

## Example 7 Simplifying an expression containing a scalar product

Expand and simplify the expression , where and are vectors.

### Solution

Expand the brackets by using property 4:

Simplify by using property 3, so:

Using property 2, simplify further to get:

## Activity 21

Expand and simplify the expression , where and are vectors.