How much of one arrow points along a given direction?
A vector is also an arrow in a plane, with a horizontal and a vertical component, (x, y). We use vectors for directions in space: for example, the direction in which a light wave oscillates, its polarization, which module 4 is all about. Its length is √(x^2 + y^2), by Pythagoras.
One question about vectors comes up again and again: how much of a vector points along a given direction? Shine a light straight down on the vector and look at its shadow on a line drawn in that direction. The length of the shadow is the projection. It equals the length of the vector times the cosine of the angle between the vector and the line.
A vector of length 1 makes an angle of 60° with a line. How long is its shadow on the line? cos 60° = 0.5. At 0° the shadow would be the whole vector, at 90° it would vanish.
Here is the link with light. A polarizer lets through only the part of the oscillation along its axis: the projection. And the energy that gets through goes as the square of that projection: cos^2 of the angle. Light polarized at 60° to a polarizer keeps 0.5^2 = 25% of its energy. This is Malus's law, which you will meet with real instruments in module 4.
Squares of lengths keep appearing: brightness from arrows, energy from projections. In quantum physics they become probabilities, as the next lesson shows.
v = (x, y), length |v| = √(x^2 + y^2).
A line at angle θ has the direction u = (cos θ, sin θ), of length 1. The projection of v on it is the dot product v · u = x cos θ + y sin θ = |v| cos α, where α is the angle between v and the line.
The share of energy along the line is (v · u)^2 / |v|^2 = cos^2 α: Malus's law.
The projection of a vector on a direction is its length times the cosine of the angle; its square gives the share of energy along that direction.