3.1 Dot Products
The dot product of two vectors u and v is a number and it is denoted by u . v. First
consider the case where the two vectors u and v are specified by a magnitude and direction. In
that case the dot product is defined by
u . v = (magnitude of u) (magnitude of projection of v on u)
= (magnitude of u) (magnitude of v) cos
= | u | | v | cos
where
 = angle between u and v
Dot products and work: Suppose an object is
acted upon by a constant force F as it moves from point P
to point Q. Then the work W done by the force on the
object as it moves from P to Q is defined by
W = (length of displacement from P to Q) 
(magnitude of projection of the force in the direction of the displacement)
__
= PQ . F
Example. Suppose a force of 3 Newtons acts on
an object as it undergoes a displacement of 3 meters and
the force makes an angle of 60 with the displacement.
Find the work done by the force on the object.
Solution.
__
__
W = PQ . F = | PQ | | F | cos
= (3 N) (2 m) cos(60)
= (3)(2)(½) Newton-meters = 3 Joules.
Work is a form of energy and in the metric system the
units of energy are Newton-meters or kilogram-meters2 / second2 which are called Joules. The
work done by a force on an object shows up in other forms of energy such as the kinetic energy
or the potential energy of the object.
 u1 
 v1 
For vectors u =  u2  and v =  v2  that are lists of numbers the dot product is defined as
 u3 
 v3 
follows.
u . v = u1v1 + u2v2 + u3v3
3.1 - 1
Proposition. Suppose u and v are vectors that are specified by a magnitude and direction
and one introduces a coordinate system so that u and v are represented by lists of numbers, i.e.
 u1 
 v1 
u =  u2  and v =  v2 . Then u . v = u1v1 + u2v2 + u3v3.
 u3 
 v3 
Proof. Apply the law of cosines to the triangle whose sides are u and v. One obtains
| v – u |2 = | u |2 + | v |2 – 2 | u | | v | cos
(v1 – u1)2 + (v2 – u2)2 + (v3 – u3)2 = (u1)2 + (u2)2 + (u3)2 + (v1)2 + (v2)2 + (v3)2 – 2 u . v
-2 (u1v1 + u2v2 + u3v3) = – 2 u . v
u . v = u1v1 + u2v2 + u3v3
Finding the angle between two
vectors. We can use the above formulas to find
the angle between two vectors that are given in
component form, i.e.
 = cos-1  | u |
u.v 
|v|
Example. Find the angle between
u = 2i – 3j + k and v = i + 2j - k.
 2
 1
Solution. u =  -3  and u =  2 , so
 1
 -1 
 = cos-1  | u |
u.v 
|v|
(2)(1) + (-3)(2) + (1)(-1)
 = cos-1  -5 
= cos-1  2
2
2
2
2
2
 2 + (-3) + 1 1 + 2 + (-1) 
 14 6
-1
= cos (-0.546) = 2.15 (radians) = 123.1
Orthogonal vectors. Two vectors u and v are orthogonal if they are perpendicular, i.e. if
the angle  between them is 90. This occurs precisely if cos = 0 or u . v = 0.
Example. Let P be the set of vectors u = xi + yj + zk that are orthogonal to the vector
v = - i + 2j + 2k. Note that P is a plane passing through the origin. Find the equation that points
on this plane satisfy.
Solution. Points on this plane satisfy
u.v=0
(xi + yj + zk) . (- i + 2j + 2k) = 0
- x + 2y + 2z = 0
Example. Let P be the set of vectors u = xi + yj + zk that satisfy the equation
3x - 2y + z = 0. Describe P geometrically.
3.1 - 2
Solution.
3x - 2y + z = 0
(xi + yj + zk) . (3i - 2j + k) = 0
u.v=0
where v = 3i - 2j + k. This is the plane consisting of all vectors u that are orthogonal to v.
Projection of one vector on another. Suppose we are given two vectors u and v and we
want to find the projection of v on u. We start with the formula
u . v = (magnitude of u) (magnitude of projection of v on u)
= | u | (magnitude of projection of v on u)
Solving for the magnitude of the projection v on u gives
u.v
magnitude of projection of v on u = | u |
In order to get the projection of v on u we have to multiply a unit vector in the direction
u
of u by this. A unit vector in the direction of u is | u |. So
Projection of v on u =
(u . v) u
| u |2
Example. Find the projection of v = 3i + 2j + 2k on u = 2i + 2j + k.
(u . v) u
(3*2 + 2*2 +2*1) (2i + 2j + k)
=
| u |2
22 + 22 + 12
12 (2i + 2j + k)
8
8
4
= 3i+3j+3k
9
Solution. Projection of v on u =
=
Algebraic properties of dot products. It is not hard to see that
u.v = v.u
(the dot product is symmetric)
2
.
u u = |u|
(cu) . v = c (u . v)
if c is a number (the dot product is homogeneous)
u . (v + w) = u . v + u . w (a distributive law)
i.i = 1
j.j = 1
k.k = 1
i.j = 0
i.k = 0
j.k = 0
3.1 - 3