The dot product
Adding and scaling vectors never says how far apart two directions are. Can one number, taken straight from the components, say it?
Where you are 3 inside
What this area is about
For \mathbf{u} = \langle a_1, b_1 \rangle and \mathbf{v} = \langle a_2, b_2 \rangle, the dot product \mathbf{u} \cdot \mathbf{v} = a_1 a_2 + b_1 b_2 is a single number, not a vector. For nonzero vectors it equals \|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta, with \theta the angle between them (0 \le \theta \le \pi). So it measures how well two directions line up, and that one fact drives everything in this area.
How these fit together
The definition and its meaning. The dot product and the angle between vectors defines the number from components and proves the cosine formula, which recovers the angle with one inverse cosine.
Two results built on it. Orthogonal vectors are those with \mathbf{u} \cdot \mathbf{v} = 0; for nonzero vectors that means exactly a right angle. Vector and scalar projection divide the dot product by \|\mathbf{v}\| or \|\mathbf{v}\|^2 (needing \mathbf{v} \ne \mathbf{0}) to measure how much of one vector runs along another and to split a vector into perpendicular pieces; the same lesson finds the work a constant force does as the dot product \mathbf{F} \cdot \mathbf{d}.
A path through it
The cosine formula comes first, since both other lessons rest on it. Take orthogonal vectors next: projection uses orthogonality to split a vector into a piece along \mathbf{v} and a piece perpendicular to it.
After this, the same product carries into higher dimensions in dot product, length and orthogonality in n dimensions, and orthogonality sets up the cross product.