Vectors

• A vector has magnitude and direction; it is written a, →OA or as a column vector.
• Multiply by a scalar to stretch (and reverse if negative); add and subtract by components.
• Magnitude of (x, y) column vector = √(x² + y²).
• Resultant: follow a route of known vectors from start to finish.
• Proofs: parallel means one vector is a scalar multiple of the other; collinear means parallel with a common point.

Magnitude and direction

A vector has both size (magnitude) and direction; a scalar has size only. The exam writes vectors as a (you write a with a line underneath) or as →OA, the vector from O to A. A column vector (x over y) means x across (right positive) and y up (positive). In these notes a column vector is written (x, y) to save space.

→AB = −→BA: the same journey reversed. Equal vectors have the same magnitude and direction, wherever they are drawn.

Work component by component

ka is parallel to a and k times as long; −a is the same length in the opposite direction. Add or subtract column vectors by adding or subtracting the top and bottom numbers separately.

Pythagoras on the components

The magnitude |a| is the length of the arrow, found by Pythagoras.

Follow a route

To find a vector between two points, go along any route made of vectors you know, adding them as you go and using a minus sign whenever you travel against an arrow.

Parallel and collinear

Two vectors are parallel if one is a scalar multiple of the other. Three points are collinear (on one straight line) if two of the vectors joining them are parallel and share a point. Write the conclusion in words.