# 3.5 Vector algebra

Throughout this course we have explored calculations involving vectors, and from these we can identify general properties of addition, subtraction and scalar multiplication of vectors. For example, using the properties of addition and scalar multiplication we found that for any vectors and :

All the algebraic properties of vectors can be summarised using eight basic algebraic properties. These are mathematical expressions of results that may seem like common sense, but when we express them using this notation, it confirms the ways that we can apply standard rules of algebraic manipulation to vectors. Notice that multiplication and division of vectors are not included in the list; this is because these operations are defined differently for vectors, and we will explore one definition of vector multiplication in the next section.

## Properties of vector algebra

The following properties hold for all vectors , and , and all scalars and .

In many ways vector quantities behave in a similar manner to scalar quantities, and these eight properties allow us to perform some operations on vector expressions in a similar way to numbers, or algebraic expressions.

## Example 6 Simplifying vector expressions

Simplify the vector expression

### Solution

Expand the brackets using property 5:

Collect like terms using property 6:

The properties of vector algebra also allow us to manipulate equations containing vectors, which are known as **vector equations**, in a similar way to ordinary equations. For example, we can add or subtract vectors on both sides of such an equation, and we can multiply or divide both sides by a non-zero scalar. We can use these methods to rearrange a vector equation to make a particular vector the subject, or to solve an equation for an unknown vector.

## Activity 18

a.Simplify the vector expression

b.Rearrange the vector equation

to express in terms of and .