An introduction to vectors
Vectors can be a strange concept when you first start using them in maths of physics but they are actually simple once you get used to them. Whereas we are used to dealing with scalars (otherwise known as numbers) which simply have a size vectors have both size and “direction”. That is vectors contain a lot more information than just numbers.
Note// Whilst when first being introduced to vectors we usually are dealing with some sort of physical space, such as a plane, vectors can actually be used much more generally for example functions actually form and infinite dimensional vector space and colours are really just an example of another 3D vector space.
To start with vectors can be thought of as “arrows” on either a plane or in space (from the more general definition of vectors you might use later you’ll find that you can use any set of “axis” or bases such as functions). The magnitude of a vector is represented by the length of the line whilst the direction is by the direction of the arrow.
Lets consider the 2D plane.
The vector a is drawn an arrow in order that we can show the direction. The length of the arrow indicated the magnitude. By reversing the the direction of the arrow we can get -a.
Equality of Vectors
Two vectors are equal if they have the same direction and size or magnitude. This means that when we represent vectors using arrows two vectors can be equal even if they are drawn in different places provided they are parallel, pointing the same way and of the same length.
Addition of Vectors in the Plane
We can also add vectors together. To do this we “join” them so that we can see the combination of where they are heading. This means that adding vectors changes both the direction and the magnitude of the vector.
Multiplication of a Vector by a Scalar
And finally we can multiply a vector by a scaler.
This only changes the magnitude of the vector. In the case in the diagram multiplying a vector by 2 doubles its magnitude but keeps the direction the same.
More complex things can be done with vectors, such as the dot product, and they can also be written down numerically so that computations can be done with them which will be shown in later lessons.