## Three Dimensional Vectors

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 .

It can also be written in terms of the basic unit vectors as p = xi + yj + zk where 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. Length of a Three Dimensional Vector

The length of a vector is called its magnitude.
If This property can be shown using Pythagoras' Theorem.  = b − a