Scalar Product
Aims: By the end of this chapter, you
will be able to
(i) calculate scalar product given two vectors, and
(ii) employ scalar product to solve for the angle between two vectors.

A scalar product is also called dot product
because the symbol that is used to represent this product of two vectors
is simply a dot •.
 It is call "scalar product" because the result
of this operation is a scalar (a number) and NOT a vector.
 The scalar product of two vectors a and b
is defined by
a
• b =  a
  b  cos θ (1) 
where θ is the angle between the wo vectors a
and b.
 Note that the angle θ is measured between the direction of these two
vectors as shown in diagrams below.
 Looking at the formula above (this is also in the formulae booklet page
5), we see that the right hand side is basically the product of three items:
the magnitude of vector a (scalar), magnitude of
vector b (scalar) and cos θ (scalar). Thus,
the product must be a scalar.
 The above formula can be expanded to vectors in three dimensions ( i,
j, k ) and beyond.
 In mechanics, work (energy) is the scalar product of Force
and Displacement. The scalar product is often used to find the angle between
two vectors.
cos θ = 
a
• b 

 a 
 b  
or simply θ = arccos [
a • b / (  a
  b  ) ]
 Example 1: Let there
be two vectors b_{1}i + b_{2}j +
b_{3}k and c_{1}i
+ c_{2}j + c_{3}k.
Then the scalar product of these two vectors is simply:
(b_{1}i
+ b_{2}j + b_{3}k)
• (c_{1}i + c_{2}j+
c_{3}k) = (b_{1}c_{1})
+ (b_{2}c_{2}) + (b_{3}c_{3}) (2) 
This method works for three and more dimensions as well. Note
that this is basically the sum of products between values belonging to the
same basis (i or j or
k).
Box 1. How to make sense of expression (2) above.
From the diagram on the left, we can let
vector b = b_{1}i
+ b_{2}j + b_{3}k
and
vector c = c_{1}i
+ c_{2}j +
c_{3}k.
Applying Cosine Rule will allow us to write
2 b   c 
cos θ =  b ^{2} + 
c ^{2}   a ^{2}
2 b   c  cos θ
= (b_{1}^{2}+b_{2}^{2}+b_{3}^{2})+(c_{1}^{2}+c_{2}^{2}+c_{3}^{2})[(b_{1}c_{1})^{2
} +(b_{2}c_{2})^{2 }
+ (b_{3}c_{3})^{2 } ]
2 b   c  cos θ
= 2b_{1}c_{1} + 2b_{2}c_{2} + 2b_{3}c_{3}
2 b   c  cos θ
= 2(b_{1}c_{1} + b_{2}c_{2} + b_{3}c_{3})
 b   c 
cos θ = (b_{1}c_{1} + b_{2}c_{2}
+ b_{3}c_{3})
Thus, we obtained the required expression (2) above. 
 Example 2: Find the scalar product of 3i
+ 5j and 7i
 3j.
Solution: (3i + 5j)
• (7i  3j) = (3x7)
+ (5x(3)) = 21 15
= 6.
 Example 3: Find the
scalar product of
( 
4 
) 

( 
1/2 
) 
8 
and 
5 
10 

12 
Solution: (4i + 8j
+ 10k) • (1/2i 
5j + 12k) = (2) + (40)
+ (120) = 82
 Example 4: Find the angle that is between
the two vectors in example 3.
Solution: θ =
arccos [ a • b
/ (  a   b
 ) ]
From example 3 above, we know the scalar product to be 82. Now, we need to
find the magnitude of these two vectors. The magnitude of the first vector
is √[ (4)^{2} + (8)^{2 } + (10)^{2} ] = √(180)
and
The magnitude of the other vector is √ [ (1/2)^{2} + (5)^{2}
+ (12)^{2}]= √(169.25).
θ =
arccos [ 82
/ ( √(180) √(169.25)
) ] ≈ 62.0^{o} ( 3 s.f) or 1.08 radian ( 3.s.f). [you
answer should be either in degree or radian]

Example 5: Let u
= u_{1}i + u_{2}j
+ u_{3}k , v = v_{1}i
+ v_{2}j + v_{3}k
and w = w_{1}i
+ w_{2}j + w_{3}k.
Show that u • (v
+ w) = u •
v + u
• w ; that is scalar product is
distributive over addition.
Solution:
LHS = (u_{1}i + u_{2}j
+ u_{3}k)
· [(v_{1}+ w_{1})i
+ (v_{2} + w_{1})j
+ (v_{3 }+ w_{1})
k]
LHS = u_{1}(v_{1}+
w_{1}) + u_{2}(v_{2}+
w_{2}) + u_{3}(v_{3}+
w_{3})
LHS = u_{1}v_{1}+ u_{1}w_{1}
+ u_{2}v_{2}+
u_{21}w_{2})
+ u_{3}v_{3}+
u_{3}w_{3}
LHS = u_{1}v_{1}+
u_{2}v_{2}+
u_{3}v_{3}+
u_{1}w_{1}
+u_{2}w_{2}+
u_{3}w_{3}
LHS = u ·
v + u
·
w
LHS = RHS
Note that this property also holds for vector in other dimensions
and over substraction.
Worksheet.
 Let vector a and vector b
be i and j respectively,
i.e a = i + 0j
and b = 0i + j.
 What is the relationship between vector a
and vector b? [hint: draw these vectors]
 What is the scalar product between these two vectors?
 What is the angle between these two vectors?
 Let vector a be i +
j and vector b be 5i
+ 5j.
 What is the relationship between vector a
and vector b? [hint: draw these vectors]
 What is the scalar product between these two vectors?
 What is the angle between these two vectors?
 Let vector v be i +
j.
 What is the magnitude of vector v?
 Find the value of  v ^{2} ?
 Calculate v •
v.
 What is the relationship between v •
v and  v ^{2}
?
 Calculate the angle between 3i
+ 5j and 7i
 3j. [Ans: 82.2^{o} ]
 Show that if a and b
are nonzero vector in three dimensions and are perpendicular
to each other then a• b = 0.
 (i) Show that if a and b
are nonzero vector in three dimensions and parallel to each
other then
a•
b =  a   b

(ii) Show that if a and b
are nonzero vector in three dimensions and antiparallel
to each other then
a• b = 
a  b 
 Let u and v be defined
as in example 5 above.
(i) Show that u• (u  v)
 v •
(u  v) = (u
 v) •
(u  v)
(ii) With the help of the diagram above and the property in (i), derive the
cosine rule.
Properties in (5) and (6) are true for vectors of other dimensions. The results
of the above questions above are important properties of scalar product. Jot
down these results in the box given below.
Important Properties of Scalar Product.
v •
w = w •
v  commutative 
u
• (v+ w) = (u• v) + (u• w)  distributive 
(k v)
• w = k
(v •
w) 
(write results from 1)

(write results from 2 here)

(write results from 3iv above here)

 Scalar product is not closed because the scalar product of two vectors is
NOT a vector but a scalar number.
 Scalar product has no identity or inverse.