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 VectorsIn three dimensions, the position vectorpcan be shown as .It can also be written in terms of the basic unit vectors as

p =xi +yj +zkwhereare 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. Skewvectors never meet.Vector

sis behind vector t.

Properties of Three Dimensional Vectors

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

kais the vectoramultiplied by the scale factork.

It is in the same direction asaandktimes the magnitude (length).e.g.

Length of a Three Dimensional VectorThe length of a vector is called its

magnitude.

IfThis property can be shown using Pythagoras' Theorem.

Addition and Subtraction of VectorsVectors 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,

addtheoppositeor inverse matrix.a − bis drawn asa + (-b).Remember this important property from two-dimensional vectors also applies in 3 dimensions.

=b −a