Vectors can also be used in three dimensions. This requires three mutually perpendicular (orthagonal) axes, as shown below.

The x- and y-axes are usually shown horizontally and the z-axes is vertical (upwards).

The position of a point can be given using three coordinates (x, y, z).

The origin O is given by (0, 0, 0).


Representation of Three Dimensional Vectors
In three dimensions, the position vector can be shown as Y11_Three_Dimensional_Vectors_02.gif.

It can also be written in terms of the basic unit vectors as p = xi + yj + zk where

Y11_Three_Dimensional_Vectors_03.gif are the basic unit vectors in the x-, y-, and z-directions.

Many of the rules of vectors that apply in two dimensions also apply in three dimensions, although a three dimensional vector cannot have a gradient. In three dimensions vectors can be parallel and never meet or non-parallel and skew. Skew vectors never meet.


Vector is behind vector t.

Properties of Three Dimensional Vectors

Multiplication of a Three Dimensional Vector by a Scalar
A vector can be multiplied by a scalar or constant. Each component of the vector is multiplied by the scalar.

The vector ka is the vector a multiplied by the scale factor k.
It is in the same direction as a and k times the magnitude (length).

e.g. Y11_Three_Dimensional_Vectors_05.gif

Length of a Three Dimensional Vector

The length of a vector is called its magnitude.
If Y11_Three_Dimensional_Vectors_06.gif

This property can be shown using Pythagoras' Theorem.

Addition and Subtraction of Vectors
Vectors can be added and subtracted in matrix form by adding or subtracting corresponding elements.


Vectors may be added using vector triangles.The second vector is added on to the end of the first vector. Note that the arrows on the two vectors being added go in one direction around the triangle and the arrow of the resultant vector goes in the other direction.

When subtracting, using a diagram, add the opposite or inverse matrix. a − b is drawn as a + (-b).

Remember this important property from two-dimensional vectors also applies in 3 dimensions.

Y11_Three_Dimensional_Vectors_08.gif = b − a