From the fundamental theorem in three dimensions, we known that any vector in space can be expressed as a linear combination of three non-zero non-coplanar vectors. In the orthogonal system of vectors we choose these vectors as three mutually perpendicular unit vectors denoted by , and directed along the positive directions of X, Y and Z axes respectively.

The initial point of any vector in space can be brought to the origin (O) (since we are dealing with free vectors) and then the terminal point can be associated uniquely with a point P(x, y, z). Now the vector becomes . Its magnitude is Corresponding, to any point P(x, y z) we can associate a vector w.r.t. a fixed orthogonal system and then this vector is the position vector (p.v.) of that point. i.e. p.v. of P =

Distance of P from O = =

x, y, z are called the components of the vector

If a vector makes angles a, b, g with the positive directions of X, Y and Z axes respectively, then cosa, cosb, cosg are called the direction cosines (d.c.’s) of .

cosa = cosb; cosg =

So that cos^{2}a + cos^{2}b + cos^{2}g = 1

Unit vector in the direction of is

= .