Chapter 7.5 – Scalar and Vector Projections
Refer to the diagrams below. The scalar projection of a on b is obtained by drawing a
line from the head of vector a perpendicular to OB or to an extension of OB . Note that
the scalar projection is zero if   90 and is negative if 90    180
Scalar Projection of a on b
The scalar projection of vector a onto b is ON, where ON  a cos
(include diagram from p. 390)
Example: Suppose that the angle between a and b is  .
a.
Determine the scalar projection of
a
on
b
b.
Determine the scalar projection of
b
on
Example 1 from page 392:
a. Show algebraically that the scalar projection of a on b is identical to the scalar
projection of a on 2 b
b. Show algebraically that the scalar projection of a on b is not identical to the
scalar projection of a on -2 b
a
Example 2 from page 392:
For the vectors a   3,4,5 3 and b   2,2,1 , calculate each of the following scalar
projections.
a.
b.
a on b
b on a


Calculating Scalar Projections

The scalar projection of a on b is
a b
b

The scalar projection of b on a is
a b
a

In general
a b

ba
b
a
Example 3 from page 394:
Determine the angle that the vector OP  2,1,4 makes with each of the coordinate axes.
Direction Cosines for OP  a, b, c 
If  ,  , and  are the angles that OP makes with the positive x-axis, y-axis and zaxis, respectively, then
cos  
a, b, c   1,0,0 
OP
a
a b c
2
2
2
; cos  
b
a b c
2
2
2
; cos  
c
a  b2  c2
2
Example 4 from page 396
For the vector OP   2 2 ,4,5 , determine the direction cosine and the corresponding
angle that this vector makes with the positive z-axis.


Examining Vector Projections
The calculation of the vector projection of a on b is simply the corresponding scalar
projection of a on b , multiplied by a unit vector in the direction of b . We know from
our earlier work that
b
is a unit vector in the direction of b .
b
Vector Projection of a on b
The vector projection of a on b
= (scalar projection of a on b )
=
a b

b
=
a b
2

(unit vector in direction of b )
b
b
b
b
 a b 
b , b  0
= 
 bb 


Example 5 from page 397
Find the vector projection of OA  4,3 on OB  4,1