A scalar is any physical quantity that has magnitude, but no direction. Having magnitude means that the quantity has a size or extent.
Examples of scalars include the word-count of a given book, the volume of liquid in a container, or the brightness of a light-bulb. In the example below you can use the to change the brightness of the light-bulb (measured in lumens).
Scalars by definition have no direction. If someone tells you they ran a 5k, they've given you no information about what direction they ran, only how far they've gone. Maybe they ran in a straight line, and so were 5km away from their start point by the time that they finished the run. However they could also have ran the 5k in a loop, so by the end of the run they ended up in the same place they started.
A sailor trying to travel from London to New York would need more than just a scalar to get there. If they were told they needed to travel 3500 miles, and then given no other information, then they could end up at any number of places that notably aren't New York.
To solve this problem, along with the distance you can also give a direction, for instance 10 miles East, or 100 metres upwards.
A physical quantity that has both magnitude and direction is known as a vector. Graphically, a vector can be represented with an arrow, where the length of the arrow represents its magnitude, and its direction represents its direction.
If I gave the sailor a vector "travel 3500 miles at a bearing of 280 degrees", they may actually be able to reach the destination.
If you have multiple vectors, and follow them like a list of instructions, then it is called adding vectors. Take the following two vectors "100 metres North" and "100 metres East". Adding the two vectors gives a new vector - "141 miles North East"
You can think of this as follows. First travel 100 metres East, then travel 100 metres North. Your sum vector is the distance and direction of your final position relative to your start position.
In a similar way, a vector can also be broken down into its component parts. In the space below you can drag your mouse/finger and it will draw a vector, as well as its horizontal and vertical components.
There are few ways of writing vectors using mathematical notation. One form of notation is called "matrix notation" and is similar to how coordinates are formatted.
You can see below how the x and y components relate to a given vector.
Another way of writing vectors is using unit vectors. A unit vector is any vector with a magnitude of one. You can add lots of unit vectors together to make longer vectors.
In A-Level there are two important unit-vectors: "i" and "j". i is drawn horizontally to the right, and j is drawn vertically to the top. With the right combination i and j you can create any 2D vector.
In the space below you can draw a vector which will be shown with unit vector notation.
Vector notation like the ones shown above make it easier to complete calculations on vectors. Adding vectors with vector notation can be completed as shown below.
You can think about this again by thinking of the vectors as a list of instructions.
Travel a metres to the right, then travel c metres to the right.
Overall you travel a + c metres to the right.
Below is an interactive version of this, and you can adjust the x and y of the
The same principle can be used to add unit vectors.
And with addition of vectors, that covers the basics of vectors and scalars as needed for A-Level Maths.
From the navigation of spacecraft to rendering graphics in videogames, vectors are an incredibly useful tool with a very simple definition - magnitude and direction. This webpage also makes use of vectors - every graphic on this page uses a file format called SVG - "Scaleable Vector Graphics".