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
![](https://koushi.in/wp-content/uploads/2022/09/representation-of-a-vector-by-coordinate.png)
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.
=
=
.
![](https://koushi.in/wp-content/uploads/2022/09/vector-coordinate-problem.png)
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 =
.