### UNIT 10 : Vector space of real numbers

**Key unit competence**Determine the magnitude and angle between two vectors and to be able to plot these vectors. Also, be able to point out the dot product of two vectors.

**Learning objectives****10.1 Vector spaces lR**^{2}**Definitions and operations on vectors**In Senior 2, we were introduced to vector and scalar quantities.

**Task 10.1**1. Define a vector.

2. Differentiate between a vector and a scalar quantity.

3. Use diagrams to illustrate equal vectors.

A quantity which has both

**magnitude**and**direction**is called a**vector**. It is usually represented by a directed line segment. In our daily life, we deal with two mathematical quantities:(a) one which has a defined magnitude but for which direction has no meaning (example: length of a piece of wire).

In physics, length, mass and speed have magnitude but no direction. Such quantities are defined as

**scalars**.(b) the other for which direction is of fundamental significance. Quantities such as force and wind velocity depend very much on the direction in which they act. These are

**vector**quantities.**Addition of vectors**Vectors are added by the triangle law or end-on rule or parallelogram law of addition.

**The parallelogram law****Vector subtraction****Multiplication of a vector by a scalar**When a vector is multiplied by a scalar (a number) its magnitude changes. If it is multiplied by a positive number the direction remains the same.

However, if it is multiplied by a negative number, the direction of the vector reverses.

**10.2 Vector spaces of plane vectors ( lR, V, +)**The definition of vector space denoted by V needs the arbitrary fields F = whose elements are called scalars.

**Definitions and operations****Properties of vectors****Linear combination of vectors****Spanning vectors****Linear dependent vectors****Linear independent vectors****Basis and dimension of a vector space****Dimension of vector space**The dimension of a non-zero vector space V is the number of vectors in a basis for V. Often we write Dim (V) for the dimension of V.

For

**lR**^{n }, one basis is the standard basis, and it has n vectors. Thus, the dimension of n is**lR**^{n}.**10.3 Euclidian vector space****Activity 10.1**In groups, carry out research to determine the similarities between vector spaces and Euclidian spaces.

**Dot product and properties****These are vectors:**