When two vectors a and b are placed tail to tail, θ is the angle between them.

For non-zero vectors a and b we define the dot product as:
a b  a b cos 

The dot product of two Cartesian vectors a  a1 ,a2  and b  b1 ,b2  is
a b  a1b1  a2b2
Vectors are used in computer animation to determine the length of a shadow
projected onto a flat surface. We use this idea to define the projection of
a vector on a vector.
The scalar projection of a on b is the length of line segment OP. (this is a
number not a vector).
Let a  OA and b  OB be any two vectors forming an angle . Let N be the
point on the line OB such that AN  OB.
A
a

O
MCV4U
B
N
b
Unit 2
We define the projection of a on b to be the vector ON . We think of this
as the shadow of a on b .
ON  k b
Assuming that 0    90 ,
ON  k b
1
In triangle AON
ON  a cos 
 a 

k b 
k 
k 
ab
b
ab
a b
2
a b
b
a b
b
2
a b
bb
a b 
Therefore, the projection of a on b is ON  
 b . The projection is a
b b 
scalar multiple of b . There is no consistent notation for the projection of a
on b but you can use a  b .
MCV4U
Unit 2
Ex: If vector a = (-3, 5, 4) and vector b = (-2, 1, 2). Calculate the scalar

projection of a


on b and b
on

a.

 a  b
proj b 

a
  3,5,4   ( 2,1,2)
proj a 
( 2)  1  2

b

a
 3,5,4   ( 2,1,2)

( 3)  5  4
2

2
2

2
2
7
5
7
38
Application to math: Find the angle a vector makes with a positive axis.
These are called the direction angles or direction cosines.
Vector Projections: this produces a vector with

a
tail at A and in the same

direction as b . (or opposite direction if the scalar projection is negative).
   


a b b
proj a     
 b  b



b
Scalar projection




Unit vector in the
direction of
vector b
  
a b
  b
b
2
MCV4U
Unit 2
2
Ex: Find the vector projection of a on b where a = (4, 3) and b = (4, -1).
  ( 4,3)  ( 4,1) 
proj a  
( 4,1)
(
4

(

1
)
)



b
2
2
2
 13 
  ( 4,1)
 17 
 52  13 
 ,

 17 17 
MCV4U
Unit 2