Vector addition and scalar multiplication
One trip follows another, or a trip is taken twice, halfway, or backward. What vector describes the result?
Where you are 3 inside
What this area is about
Vectors can be combined. Two of them add tip to tail, and a vector can be multiplied by a real number, a scalar, to stretch it, shrink it, or reverse it. Both operations have a picture with arrows and a calculation on components, and the two always agree: for \mathbf{u} = \langle a, b \rangle and \mathbf{v} = \langle c, d \rangle, \mathbf{u} + \mathbf{v} = \langle a + c, b + d \rangle and k\mathbf{u} = \langle ka, kb \rangle.
How these fit together
Two definitions. Adding and subtracting vectors defines the sum by the triangle rule and the difference as the arrow that joins two heads. Scaling a vector defines k\mathbf{v}: the length is multiplied by |k|, and the direction reverses when k < 0. Each lesson refers to the other (the difference uses the opposite -\mathbf{v}), so they are best taken side by side.
An application. Forces and velocities as vectors adds velocities, such as a plane’s air velocity and the wind, and adds several forces acting on one object.
A path through it
Addition and scaling together, then forces and velocities, which need the triangle rule.
After this come the dot product, adding and scaling vectors in n dimensions, and motion in the plane.