vector

374 Vectors
OC
=
m n
Cor. If m = n = 1, then C will be the mid-point, and
mb na
OC
a b
=
2
5.7 PRODUCT OF TWO VECTORS
The product of two vectors results in two different ways, the one is a number and
the other is vector. So, there are two types of product of two vectors, namely scalar
product and vector product. They are written as
a .
b and a
b .
5.8 SCALAR, OR DOT PRODUCT
The scalar, or dot product of two vectors a and b is defined to be
a
scalar where is the angle between a and b .
Symbolically,
a .
b = a b cos
b cos i.e.,
Due to a dot between a and
b this product is also called dot product.
The scalar product is commutative
To Prove.
Proof.
a . b = b . a
b . a = b a
cos ( )
b
B
=
a b cos
0
A
a
= a .
b Proved.
Geometrical interpretation. The scalar product of two vectors is the product of one
vector and the length of the projection of the other in the direction of the first.
Let
then
=
OA
a and OB b
a . b = (OA) . (OB) cos
ON
= OA . OB . OB
= OA . ON
= (Length of a ) (projection of b along a )
5.9 USEFUL RESULTS
^
^
i . i = (1) (1) cos 0° = 1 Similarly, ^ j .
^
j = 1,
^
^
^
^
k . k = 1
i . j = (1) (1) cos 90° = 0 Similarly, ^ j . k ^
= 0, k . i = 0
Note. If the dot product of two vectors is zero then vectors are prependicular to each other.
5.10 WORK DONE AS A SCALAR PRODUCT
If a constant force F acting on a particle displaces it from A to B then,
Work done = (component of F along AB). Displacement
= F cos . AB
=
F . AB
Work done = Force . Displacement
0
A
^
b
^
a
N
B
F
A
B