The idea

Vectors in three dimensions

Extending vectors to 3D — the i, j, k and column forms, magnitude by Pythagoras in three dimensions, the distance between two points, unit vectors, and the test for two vectors being parallel.

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

Pure · Vectors

Vectors in three dimensions

Extending vectors to 3D — the i, j, k and column forms, magnitude by Pythagoras in three dimensions, the distance between two points, unit vectors, and the test for two vectors being parallel.

Why it works

One more component

A point in space needs three numbers, so we add a third axis

z

perpendicular to both

x

and

y

. Every rule you already know for 2D vectors carries over unchanged — you just carry one more component:

\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}.

Addition, subtraction and multiplication by a scalar are still done component by component. Nothing new is being defined; the dimension is the only change.

Keep reading — free

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

Start free

Takes a minute — no card.