Vectors
A segment says how far apart two points are. How do you also say which way you went, and turn that into numbers you can compute with?
Where you are 4 inside
What this area is about
A vector carries a length and a direction. Every arrow that makes the same trip is the same vector, wherever it is drawn. This area builds that idea step by step: an arrow, then a pair of numbers \langle a, b \rangle, then the length and direction read back off those numbers, then a second way of writing the same vector.
How these fit together
Definitions. Directed line segments put an arrow on a segment, so it has a start, a finish and a direction. Component form names a vector by its change across and up, \langle x_2 - x_1, y_2 - y_1 \rangle, terminal minus initial. Unit vectors and i, j notation writes \langle a, b \rangle as a\mathbf{i} + b\mathbf{j} and builds the unit vector \displaystyle \frac{\mathbf{v}}{\|\mathbf{v}\|} for \mathbf{v} \ne \mathbf{0}.
A procedure. Magnitude and direction finds \displaystyle \|\mathbf{v}\| = \sqrt{a^2 + b^2} and the direction angle of a nonzero vector, and goes back from length and angle to components.
A path through it
Directed segments, then component form, then magnitude and direction, then unit vectors: each needs the one before.
This opens vector addition and scalar multiplication, and with them the dot product, converting between polar and rectangular coordinates and vectors in space.