3D Vector Algebra

                                         3-D CO–ORDINATE SYSTEM

Cartesian Co-ordinates: -  “The theory of vectors is associated very closely with co-ordinate geometry so we shall start by introducing the co-ordinate system".

Co ordinate System

We use a rectangular Cartesian co-ordinate system with axes O_xyz or O_{x1 x2 x3}, as shown below. The position of any point, P say, is given by co-ordinates, or components       (x, y, z) or (x₁, x₂, x₃). 

The order and direction of axes are assumed to be 'right handed'. The distance from point O to point P is r, and r can be calculated using the Pythagoras theorem to give

 Direction Cosines: -

                                   We will see later that it is useful to know the angle the line OP makes with each axis.Fortunately this can be easily calculated. The shaded areas of the figures below each form right-angled triangles with the axes.

Direction Cosines

Knowing the length and the appropriate co-ordinate, each angle can be calculated.

The angle ∠POX is given by

Value of Direction Cosines

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