Vector Basics

A vector is a way to represent a movement between two points. A straight line with an arrow at the end is very often used to visualise vectors. For example vector a in the grid below shows a small upwards movement.

…vectors b on the following to grid shows a large movement to the left. The arrow on the line shows the direction of the movement.

We also use a system called the matrix system to describe vectors.

For example the grid below shows the vector c.

We can see that the vector goes 3 units to the right and 4 units up. That must mean that the vector is;

Below we shall use vector labels for the vectors.

This section explores multiplying vectors. The vector a with matrix;

…is shown on the grid below.

Suppose we wanted to multiply the vector above by 2 so that it becomes 2a. On the grid below is also the 2a vector.

…when

Suppose we wanted to half it instead of multiplying by 2. On the grid below is also the ½a vector.

…when;

Suppose we wanted to make the original vector negative. That is to make vector a vector –a. The vector a and vector –a is shown on the graph below.

A negative vector goes in the opposite direction of a positive vector. That is when…

Above in the multiplication of vectors we saw that it is obvious that adding vectors is possible because;

On the grid below is the two vectors a and b.

An insect going to move through vectors a+b. The insect moves through vector a first.

At the endpoint the insect can’t move to the start of b without moving through another vector. So what we do is assume that vector b is at the end of vector a.

Now the insect can move through vectors a+b as shown above. The insect has travelled a vector of;

But notice below that it could have travelled in the vector blue shown below. This is called the resultant vector.

So we have;

Since adding is possible that must mean that subtracting is possible. Below is the two vectors a and b. This time the insect is going to move through vectors a-b.

The vector moves through vector a first vector a is;

We have a-b, we imagine that this is a+-b. We flip the vector b as shown below such that it moves in the negative direction of b.

The insect could have travelled in the blue line, This is called the resultant vector.

So we have;