Definition of unit vector , Rectangular components of a vector , Direction cosines of a vector , Illustration on Unit vectors

1. Physics Helpline
L K Satapathy
Theory of Vectors 3

2. Physics Helpline
L K Satapathy Theory of Vectors - 3
Unit vector :
The unit vector in the direction of vector A is defined as
A =
 A 
A
 A =  A  . A
 Given Vector = ( Its magnitude )  ( Unit vector in its direction )
Unit vectors in the direction of x , y and z axes = i , j and k
(4N) i  A force of 4N is acting parallel to the x-axis
(6N) j  A force of 6N is acting parallel to the y-axis
(8N) k  A force of 8N is acting parallel to the z-axis

3. Physics Helpline
L K Satapathy Theory of Vectors - 3
Rectangular components of a vector
O
P
A
B
C
D
E
F
X
Y
Z
Given vector OP = R
Rectangular components
OA = BD = EP = CF = Rx = Rx i
OB = AD = FP = CE = Ry = Ry j
OC = AF = DP = BE = Rz = Rz k
Its scalar projections
OA along X-axis
OB along Y-axis
OC along Z-axis

4. Physics Helpline
L K Satapathy Theory of Vectors - 3
Using Polygon Law, we get
OP = OA + AD + DP
 R = Rx i + Ry j + Rz k
OA  AD O A
B
C
D
E
F
X
Y
Z
P
 OD  DP  OD + DP = OP . . . (2)2 2 2
 OA + AD = OD . . . (1)
2 2 2
(1) & (2)  OA + AD + DP = OP2 2 22
 R = Rx + Ry + Rz . . . (3)
2 2 2 2
OD in XY plane , DP parallel to z-axis

5. Physics Helpline
L K Satapathy Theory of Vectors - 3
Direction Cosines
In  OAP ,
In  OBP ,
In  OCP ,
cos  =
OP
OA
R
Rx
=
cos  =
OP
OB
R
Ry
=
cos  =
OP
OC
R
Rz
=
cos  + cos  + cos  =2 2 2 Rx + Ry + Rz
R
2 2 2
2
R
R
2
2
= = 1
 OAP =  OBP =  OCP = 90
A
B
C
D
E
F

P
O



6. Physics Helpline
L K Satapathy Theory of Vectors - 3
Illustration
Q. Find the unit vector in the direction of the vector R = 3 i + 4 j + 12 k
The given vector is R = 3 i + 4 j + 12 k
Comparing with R = Rx i + Ry j + Rz k
we get Rx = 3 Ry = 4 Rz = 12
= R =
 R 
R
( 3 i + 4 j + 12 k )
13
1
Ans:
 R = Rx + Ry + Rz = 9 +16 + 144 = 169   R  = 13
2 2 2 2

7. Physics Helpline
L K Satapathy
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