This is a vector:

A vector has a **magnitude** (size) and **direction**:

The length of the line shows its magnitude and the arrowhead points in the direction.

We can add two vectors by joining them head-to-tail:

And it doesn’t matter which order we add them, we get the same result:

### Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.

The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.

If you watched the plane from the ground it would seem to be slipping sideways a little.

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seems to fly sideways. Vectors help explain that.

Velocity, acceleration, force and many other things are vectors.

## Subtracting

We can also subtract one vector from another:

- first, we reverse the direction of the vector we want to subtract,
- then add them as usual:

**a** − **b**

## Notation

A vector is often written in **bold**, like **a** or **b**.

A vector can also be written as the letters of its head and tail with an arrow above it, like this: |

## Calculations

Now … how do we do the calculations?

The most common way is to first break up vectors into x and y parts, like this:

The vector **a** is broken up into

the two vectors **a _{x}** and

**a**

_{y}

## Adding Vectors

We can then add vectors by **adding the x parts** and **adding the y parts**:

The vector (8,13) and the vector (26,7) add up to the vector (34,20)

### Example: add the vectors **a** = (8,13) and **b** = (26,7)

**c** = **a** + **b**

**c** = (8,13) + (26,7) = (8+26,13+7) = (34,20)

When we break up a vector like that, each part is called a **component**.

## Subtracting Vectors

To subtract, first reverse the vector we want to subtract, then add.

### Example: subtract **k** = (4,5) from **v** = (12,2)

**a** = **v** + −**k**

**a** = (12,2) + −(4,5) = (12,2) + (−4,−5) = (12−4,2−5) = (8,−3)

## Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

|**a**|

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

||**a**||

We use Pythagoras’ theorem to calculate it:

|**a**| = √( x^{2} + y^{2} )

### Example: what is the magnitude of the vector **b** = (6,8) ?

|**b**| = √( 6^{2} + 8^{2}^{ }) = √( 36+64^{ }) = √100 = 10

A vector with magnitude 1 is called a Unit Vector.

## Vector vs Scalar

A **scalar** has a **magnitude** (size) **only**.

Scalar: just a number (like 7 or −0.32) … definitely not a vector.

A **vector** has **magnitude and direction** and is often written in **bold**, so we know it is not a scalar:

- so
**c**is a vector, it has magnitude and direction - but c is just a value, like 3 or 12.4