Adding and Scaling Vectors

A vector is an arrow with a length and a direction, written as components like (a, b) meaning a across and b up. To add two vectors, add their components separately: (a, b) + (c, d) = (a + c, b + d). Picture it by placing the tail of the second arrow at the head of the first — the sum is the new arrow running from the very start to the very end. Flip the order and you get the same endpoint, which is why the two arrows together with their sum sketch out a parallelogram.

Scaling a vector by a number k, written k(a, b) = (ka, kb), stretches or shrinks the arrow by a factor of |k|. If k is negative, the arrow also flips to point the opposite way. A swimmer swimming straight across a river has a velocity vector pointing straight at the far bank, but the river current adds its own vector pointing downstream. Add those two vectors tip-to-tail and the swimmer's actual path is the diagonal sum — straight across plus downstream drift lands them downstream of their target.

This is the algebra underneath everything else in the unit. Parametric motion describes a point's position as vectors changing with time, and velocity itself is a vector you'll be adding to position. Matrices later in the unit act on vectors by transforming their components all at once, so being fluent with component addition and scaling now makes that jump much easier.