Vectors

El método, paso a paso

1. 1

   Read a column vector as a movement

   (3, −2) means 3 right and 2 down

   The top number is horizontal and the bottom vertical, with negatives meaning left and down. A vector is a movement rather than a position — the same vector applies wherever it starts, which is what distinguishes it from a coordinate.

2. 2

   Add vectors by adding the components

   (3, −2) + (1, 5) = (4, 3)

   Add the tops and add the bottoms. Physically this is doing one movement then the other, and the single vector that results is the direct route between the same start and end points.

3. 3

   Multiply by a scalar to stretch

   3 × (2, −1) = (6, −3)

   Every component is multiplied, so the vector points the same way but is three times as long. A negative scalar reverses the direction as well as scaling it.

4. 4

   Subtract by adding the reverse

   AB = b − a, going from A to B via the origin

   Travelling from A to B means going backwards along a and forwards along b. This is the single most useful fact in geometric vector questions, and writing it as 'end minus start' makes the direction unambiguous.

5. 5

   Use scalar multiples to prove things are parallel

   if XY = 2 × PQ then XY is parallel to PQ

   Two vectors are parallel exactly when one is a scalar multiple of the other. If they also share a point, the three points must lie on a straight line — which is how collinearity is proved.

A vector is not a point

The notation is nearly identical, which causes real confusion. A coordinate (3, 5) is a fixed location; a vector (3, 5) is an instruction to move three right and five up, and it means the same thing wherever you start from.

Keeping this distinction alive matters because the two are combined constantly: a position vector describes where a point is by giving the movement from the origin to it, and questions slide between the two ideas without warning.

• Vector = magnitude and direction
• Add or subtract componentwise
• Scalar multiple → same direction, different length
• AB = b − a (end minus start)
• Parallel ⟺ one is a scalar multiple of the other

Proving with vectors

The higher-tier questions ask students to express a route in terms of two given vectors and then to conclude something — that two lines are parallel, or that three points are collinear, or that a point is a midpoint. The proof always rests on finding a scalar multiple.

The method is to express both routes in terms of the same two base vectors, then look for one expression being a multiple of the other. Students who try to reason from the diagram rather than from the algebra usually stall.

Getting from one point to another

Any route between two points can be built from the vectors you have been given, travelling along them forwards or backwards. Going backwards along a vector means subtracting it, which is where sign errors enter.

Drawing the route on the diagram with arrows before writing anything algebraic is the reliable method. Each arrow becomes a term, and its direction decides the sign.

How we teach vectors

We teach 'end minus start' as a fixed phrase for AB = b − a, because the reversed version is the topic's main error and it is a memory problem rather than an understanding problem.

For the proof questions we insist the route is drawn with arrows before any algebra is attempted. The algebra is straightforward once the route exists; finding the route by staring at the diagram is not.