# 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:

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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 |
If we just name the whole vector using one letter,
then we can describe it using the letter with a |

### 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.