Orthogonal system of vectors
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 cos2a + cos2b + cos2g = 1
Unit
vector in the direction of
is

=
.