Accelerated Math III
Vector and Scalar
Projections
Dot Products
If vector v has magnitude 10 and vector u
has magnitude 14 and the standard
position angle between them is 120◦,
write v u .
●
Component Review
If vector v has magnitude 10 and
standard position angle of 120◦, write v
in component form.
Components?
Suppose you want to break a vector
down into components, but the
components are not in the x and y
directions...
Components?
For example, a car sits on an inclined
plane. What part of the weight of this car
acts on the car to make it roll downhill?
Projections – from the Geometric
Perspective
A vector projection is the component of
a vector in a given direction.
Projections – from the Geometric
Perspective
The length of the vector projection is
called the scalar projection and is
|a|cosθ.
a
θ
Vector projection
b
Scalar Projections
• A scalar projection


of u onto v is the
length of the shadow
that

u

u casts on v.
v
Scalar projection
Vector Projections
The vector projection is the actual
vector component. Therefore it is a
vector with the same direction as the
second vector, but with the length of
the scalar projection.
Vector Projections
• A vector projection


of u onto v is the
vector in the same
direction as v,
whose length is the
scalar projection of
 
u on v.
u
v
Vector Projection
Orthagonal Component
The orthagonal component is the
actual vector component perpendicular
to the vector projection.
Vector Projections
• A vector component of


u orthogonal to v is the
vector perpendicular to
v, that adds to the
vector projection of
 
u on v to create u.
u
Orthogonal
Component
v
Vector Projection
Orthagonal Component
One way to find the orthagonal
component is to subtract the vector
projection from the original vector.
Why will that work?
• Now you try these!!
Think about it…
Why doesn’t the
scalar projection
depend on the length
of the second
vector??
a
θ
b
So try these!
• Find the scalar
projection of r on s
if |r| = 10, |s| = 16,
and the angle
between them is
140º.
• 10cos 140º = -7.66
•Find the length of
the orthogonal
component of r on s.
•(-7.66)2 + |o|2 = 102
•So |o| = 6.428
So What Happens If We Have
Coordinates?
• How long is the scalar projection of
u onto v?
• Shouldn’t it be:
|u|cos 
if  is the angle between vectors
• So isn’t that |u| (u • v) = (u • v)
|u||v|
|v|
Try This One!
• Find the scalar
projection, the vector
projection, and the
orthogonal component
of:
(6i + 7j ) onto (5i - 12j )
The scalar projection is:
• 6·5 + 7·(-12) = -54/13
The vector projection is:
• -54(5i - 12j )/132 =
-270i/169 + 648j/169
The orthogonal component is:
(6i + 7j ) (-270i/169 + 648j/169) =
1284i/169- 535j/169
Vector Projections
 
 
b
a

b
b
 
a cos     a
b
a b b
 
 
 
a b  b 

b b
 
a
θ
b