The idea

Column vectors

A vector is a translation — so far across, so far up — not a place. How to add, subtract and scale columns, why a scalar multiple is parallel, how Pythagoras gives the magnitude, and why the journey from A to B is destination minus start.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

Geometry & measures · Vectors

Column vectors

Why it works

A move, not a place

A vector is not a place. It is a move. The column vector

\binom{3}{-2}

is one instruction: go $3$ across, then $2$ down. The top number is the movement in the

x

direction (positive means right, negative means left) and the bottom number is the movement in the

y

direction (positive means up, negative means down). That is the entire definition, and everything else in this topic falls out of it.

A vector has no starting point

This is the part students find strangest, so make it concrete. Start at the origin and obey

\binom{3}{-2}

: you land on

(3, -2)

. Start at

(-4, 1)

and obey the same instruction: you land on

(-1, -1)

. Start at

(0, 4)

: you land on

(3, 2)

. Three different journeys ending in three different places — but one single vector, because "3 across, 2 down" is the same instruction every time.

Keep reading — free

The rest of the explanation, plus 3 worked examples you step through move by move.

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Takes a minute — no card.