ME 2560 Statics
The Dot (or Scalar) Product of Vectors
Geometric Definition
The dot (or scalar) product of two vectors is defines as follows:
A  B  A B cos( )
Here,  represents the angle between the two vectors. If one of the vectors is a unit
vector, then the dot product is the projection of the vector in the direction of the unit
vector.
A  n  A n cos( )  A cos( )
The components of A parallel and perpendicular to n are
A   A  n n
A  A  A
and
The dot product of two vectors is zero if they are perpendicular to each other.
Calculation
Given two vectors A and B expressed in terms of a mutually perpendicular set of unit
vectors i , j , and k , we calculate the dot product as



A  B  ax i  a y j  a z k  bx i  by j  bz k  a xbx  a yby  a zbz
Properties of the Dot Product
o Product is commutative: A  B  B  A
o Product is distributive over addition: A   B  C    A  B    A  C 
o Multiplication by a scalar  :   A  B    A  B  A   B 
Kamman – ME 2560: page 1/2
Example 1:
Given: Two vectors, A  10 i  2 j  8 k and B  3 i  7 j  5 k
Find: The angle between the two vectors,  .
Solution:
We can calculate the angle using the inverse cosine function
 A B 
30  14  40
1 

cos


2
2
2
2
2
2
 10  2  8 3  7  5
 A B
 44.6548  44.7 (deg)
  cos 1 

84 
1 
  cos 

 118.085 

Example 2:
Given: A vector, A  10 i  2 j  8 k , and a unit vector n   72  i   76  j   73  k .
Find: a)  the angle between the two vectors, b) A the component of A parallel to n ,
and c) A  the component of A perpendicular to n .
Solution:
a) We can calculate the angle using the inverse cosine function as before.
6
3
2
 A n 
 8 
1  (10  7 )  (2  7 )  (8  7 ) 
  cos 
 cos 1 
  cos 


 12.9615 
102  22  82


 A 
 51.8871  51.9 (deg)
1
b) A  ( A  n) n  8  72  i   76  j   73  k   2.29 i  6.86 j  3.43 k

 

c) A  A  A  10 i  2 j  8 k  2.29 i  6.86 j  3.43 k  7.71 i  4.86 j  4.57 k
Check:


A  A  2.29 i  6.86 j  3.43 k  7.71 i  4.86 j  4.57 k

 17.6327  33.3061  15.6735  0
Kamman – ME 2560: page 2/2