## Vectors (O level Maths)

This tutorial is about O level vectors for Zimsec and Cambridge students. First we are going to define a vector, then represent it on a Cartesian plane and finally calculate problems involving vectors.

## Vector representation

A vector is a quantity that has both magnitude (size, length or modulus) an direction. On a Cartesian plane, a vector is represented as a arrow joining two points.

The diagram above shows a vector which represents the displacement from point A(1; 2) to point B(5; 3).

A vector can be written in many ways:

- either , ,
- or , ,

The values of a vector are usually given as a column matrix. On the diagram above, there are dashed lines showing the horizontal displacement (in the x-axis) and vertical displacement (in the y-axis) of the vector. From the diagram we can see from the dashed lines that x = 4 units and y = 1 unit. In column form, the vector is represented as:

### Negative vectors

When dealing with vectors, direction is very important. For example, and are not the same because they are going in opposite directions. is negative of .

This means that if

Then

Which therefore means = –

### Magnitude

The magnitude of a vector (also known as the modulus) is the length or size of the vector.

Magnitude of is usually shortened to ||.

Magnitude =

Using our earlier example, || =

(in surd form)

*The modulus of a vector is always positive.*

Now let us find the modulus of .

|| =

### Scalar multiplication

If vector is multiplied a scalar k, whereby k is any number, the resulting vector is a vector k times as big as and parallel to .

For example, vector 5 is 5 times as big as vector and also parallel to vector .

### Addition and Subtraction

Vectors can be added and subtracted as follows:

If and

then

and

## Position vectors

A position vector is a vector that is tied to the origin. Let us illustrate it using a diagram.

If we have a point A, the position vector of point A is . By the same token, the position vector of point B is .

Now if a point has the coordinates (x; y), its position vector is .

Therefore, using the diagram above we can see that:

- the position vector of A = since A has the coordinates (1; 2)
- the position vector of B = since B has the coordinates (5; 3)

Now here is the thing about vectors, a displacement of followed by is equivalent to a resultant displacement of .

Therefore

## Examples

### Question 1

Given points A(7; 8) and B(2; 1), find:

a)

b)

### Solution 1

To make it easy when solving vectors you first convert all the given points to position vectors: and

a)

b)

### Question 2

The points O, P, Q, R, S have coordinates (0; 0), (1; 5), (3; 8), (7; 10), (10; 3) respectively. Express each of the following as a column vector.

a)

b)

c)

d)

e)

f)

### Solution 2

First convert the points to position vectors

a)

b)

c)

d)

e)

f)