Vectors

A scalar has size only: mass, speed, distance. A vector has size and direction: force, velocity, displacement. We draw a vector as an arrow; its length is the size and the way it points is the direction. The arrow AB points from A (tail) to B (head). Reversing it gives the negative: BA = −AB.

Write a vector as a column (x, y): x is steps across, y is steps up. To add two vectors, add the x's and add the y's. To subtract, subtract them. Geometrically, adding means putting the tail of the second arrow on the head of the first; the resultant runs from the very start to the very end.

The magnitude |a| is just how long the arrow is. Because the components form a right-angled triangle, Pythagoras gives |a| = root(x² + y²). For example (3, 4) has length root(9 + 16) = 5. A unit vector has length 1 and shows direction only; you get it by dividing a vector by its own magnitude.

The position vector OP locates a point P from the origin O. The big rule is AB = OB − OA = b − a: walk back to O, then out to B. The midpoint of AB has position vector (a + b)/2. To prove points are collinear (on one line), show one vector is a scalar multiple of another that shares a point.