Describing translations using vectors

Vectors are used to describe movement. We often use them to describe the movement of shapes on or off of graphs.

Vectors, figure 1

We also write vectors like this

Vectors, figure 2

which refers to vector AB, this can be a line or two points, point A and point B,. It can also be referred to as a or as below:

Vectors, figure 3

Adding Vectors

When it comes to adding and subtracting vectors its quite simple, add the x values and add the y values together.

Eg. Adding:

Vectors, figure 1

When vectors are added together the arrows follow the same direction eg.

Vectors, figure 2

Vectors, figure 3

Subtracting Vectors

Eg. Subtracting:

Vectors, figure 1

a-b =

Vectors, figure 2

Multiplying by a scalar

Vectors have a direction and a magnitude. So for example the vector3a is three times as large asa.

To multiply by a scalar, simply multiply the column vector by the scale factor, or multiply the length by the scalar.

For example:

Vectors, figure 1

Diagramatic Vectors

Often, we have vectors put together to describe a shape or a diagram.

For example:

Vectors, figure 1

In this diagram we can describe the different lengths in terms of their vectors.

Eg. OP=a PQ=b QR=-a RO=-b and RP =-b + a

You may often need to find vectors that describe how to get from one point of the diagram to the other.

For example:

Vectors, figure 2

Given that:

Vectors, figure 3

And point C is on the line DE such that EC: DC is in the ratio 1:3 find an expression for the line the vector

Vectors, figure 4.

Firstly we need to think about the line DE and how we can express that as a vector.

DE =b-a

Now we need to think about the ratio. We know that the line DE is split in the ratio of 1:3. Thereforeb-a = the whole. When we have a ratio we need to split a whole into equal parts, in this case 1 part to 3 parts.



1/4 (b-a) : 3/4 (b-a)

Now we can work out the whole vector

Vectors, figure 5

To go from F to D =a

Then we will need to add the vector

Vectors, figure 6

which we know is 3/4 (b-a)

a + 3/4 (b-a) =

a + 3/4 b- 3/4 a = 1/4 a +1/4 b

Vectors for proofs

Often you will be required to prove a vector can be written in a certain way or that one vector relates to another in a certain way.

Common things that you may need to prove:

  1. That two lines are parallel
  2. That a point on a line is a midpoint.

For these specific proofs remember

  1. When two lines are parallel they have the same gradient. Therefore if two vectors have a different magnitude but the same letter eg. 4v and2v they are parallel.
  2. A midpoint is half of a line, so the scalar of the vector should be ½ of the given vector.

For example to prove that VY is parallel to ZX , given:

Vectors, figure 1

Firstly, put all the information you have about the vectors onto the diagram.

Vectors, figure 2

To prove that the two lines are parallel we are going to need to find the vectors for both lines.

So, let’s start with the line ZX

We can get from Z to X by adding the vectors

Vectors, figure 3

= a + 3b + -a+b

= 4b

Now we just need to work out the vector for the line VY

Which we can get to by adding the vectors

Vectors, figure 4

= 1/3(a+3b) + -1/3 a

= 1/3 a +b - 1/3 a

= b

Therefore because b is a factor of 4b the lines must be parallel!