# Vectors

Scalars are things which just have a value – for instance ‘5’ is a scalar quantity – all it has is a value of ‘5’, it doesn’t have any other characteristics or properties.

Vectors have both a magnitude value and a direction – so they’re made up of two things.  An everyday example of a vector quantity is if you are describing the movement of a ship or plane.  You might say something like, “the plane travelled 2000 km north.” You could represent this movement on a diagram by an arrow, like this:

This is a vector quantity because it has a magnitude (2000 km) as well as a direction (north).

There are two main ways to talk about a vector – you can either describe the two points the vector points between, or just name the whole vector one thing:

 In this case, we can describe this vector as  or . The arrow above the AB or the tilde sign below the AB tells us that it’s a vector. If we just name the whole vector using one letter, then we can describe it using the letter with a tilde sign underneath it to tell us it’s a vector, like this:.

### Vector magnitudes

Sometimes we want to talk about only one of the two parts a vector is made up of, so only one out of the magnitude or the direction.  A common example of this is when we’re talking about how far someone might have walked – we don’t necessarily care which direction they walked, but just how far it was.  Say we had a person who walked 15 km in the direction west.  The vector showing their walk would be:

To talk just about the magnitude of this vector, but not the direction, what you can do is write the vector letter down and put a pair of vertical lines around it:

When we talk about the entire vector, then it would look something like this:

Notice how now we’ve described not only the distance but the direction of travel as well.

### Equal vectors

For normal numbers to be equal, they have to have the same value.  For instance, .  For vectors to be equal, they have to have the same magnitude and the same direction:

### Taking the negative of a vector

If you put a negative sign in front of a vector, you reverse its direction.  It keeps the same magnitude however.  So if I have:

Putting a negative sign in front of my vector makes it reverse direction.  So if I wanted to draw , I’d get:

### Multiplying a vector by a scalar

If you want to change the magnitude but not the direction of a vector, you can multiply it by a scalar.  Any scalar larger than 1 will make the vector longer, any scalar smaller than 1 will make it shorter.  For instance:

The  vector has a magnitude of 3.  When we multiply it by a scalar value of 2, the magnitude doubles in size, but the direction remains the same.  So we get a new vector, , with a magnitude of 6, but with the same direction as the original vector.