Representation of a vector by coordinates:
Let us consider OX, OY and OZ are three perpendicular axes, where O is the origin. Let P is a point with coordinates (x, y, z) and 
= 
is the position vector of P. 
, 
and 
are the unit vectors along + ve X, Y and Z axis. Let PB is the perpendicular drawn on x-y plane. BA and BC are the perpendicular drawn on X and Y axis respectively. PD is the perpendicular drawn on Z axis.
Therefore 
= x, 
= 
= y and 
= 
= z and their magnitudes are OA = x, AB = OC = y and BP = OD = z.
Or, = + +
= x + y + z.
Magnitude: From 
OAB, OB2 = OA2 + AB2 and from OBP, OP2 = OB2 + BP2
From OAB, 
= + and from OBP, = +
Or, OP2 = OA2 + AB2 + BP2
So, R2 = x2 + y2 + z2
R = 
Direction: Let us consider 
, 
and 
are the angles of R with +ve X, Y and Z axis respectively. Then, cos = 
, cos = 
, cos = 
where cos, cos and cos are the direction cosines of .
Example: 1. The coordinates of two points P and Q are respectively (
, 
, 
) and (
, 
, 
) then find 
.
The position vector of point P is 
= î + ĵ + k̂ and the position vector of point Q is 
= î + ĵ + k̂.
Using triangle law of vector addition 
= –
Or, = (î + ĵ + k̂) – (î + ĵ + k̂)
= ( – )î + ( – )ĵ + ( – )k̂.
= 
= 
.
2. If 
(= 2î + nĵ – k̂) creates angle 
with positive y axis then what is the value of n?
Direction cosine cos
= 
or, cos = 
Or, 
=
n = 
.