Vectors in the plane
A vector is an arrow in the plane (later we'll deal with vectors in higher-dimensional spaces). We encode this arrow as a pair of numbers . The number tells us how far to the right the arrow points; the number tells us how far upwards it points. If the arrow points to the left then is negative; if it points downwards then is negative.
Can you match up the vectors with those in the diagram? (Some of the vectors are not depicted).
(answers at ).
A lot of this module will focus on the interplay between algebra (like column vectors) and geometry (like arrows in the plane).
Suppose is a vector. What is it's length? By Pythagoras's theorem, it's . I'll write this as , which you can read out loud as "norm ". The angle that makes with the horizontal is (by trigonometry). If we want to write and in terms of and , we get (again, using trigonometry): So .
What happens if I rotate by an angle anticlockwise? We get a new vector , which we can express in terms of and .
Rotation preserves lengths, so .
The angle that makes with the horizontal is . Therefore
We can expand this using the trigonometric addition formulae:
Using and , we get
This expresses in terms of the original vector and the angle of rotation. We now invent a piece of notation which separates out the dependence of on from its dependence on : we write You can just think of this as a shorthand for , keeping track of where all the coefficients sit.
More generally, given a 2-by-2 array of numbers and a vector , we define the product This defines the action of a matrix on a vector. This notation completely separates out the rotation ( ) from the vector we started with ( ).
Now we don't have to limit ourselves to rotations: any matrix defines a geometric transformation of the plane. This is the transformation We'll see lots of examples in the next video (rotations, reflections, shears,...).
How do you remember the formula for a matrix acting on a vector? The mnemonic I like is as follows. To get the first entry of , you ``multiply the top row of into '', that is you perform the multiplications and (working across the top row of and down the column of ) and sum them.
To get the second entry, you multiply the second row of into .
In the next video, we'll see lots of examples of transformations of the plane coming from 2-by-2 matrices.