Dot Product
So far, we haven’t talked about how to
multiply two vectors…because there are two
ways to “multiply” them.
Def. Let a  a1 , a2 , a3 and b  b1 , b2 , b3 ,
then the dot product is
a  b  a1b1  a2b2  a3b3
This is also called the scalar product, since
the result is a scalar
Ex. Find the dot product
a) 2, 4  3, 1
b) (i + 2j – 3k) ∙ (2j – k)
Properties of the Dot Product
1) a ∙ a = |a|2
2) a ∙ b = b ∙ a
3) a ∙ (b + c) = a ∙ b + a ∙ c
4) (ca) ∙ b = c(a ∙ b) = a ∙ (cb)
5) 0 ∙ a = 0
Dot product is used to find the angle between
two vectors:
Thm. If θ is the angle between a and b, then
a  b  a b cos
Ex. Find the angle between a  2, 2, 1 and
b  5, 3, 2
Ex. Consider the points A(3,1), B(-2,3) and
C(-1,-3). Find mABC.
Thm. Two vectors a and b are orthogonal if
a ∙ b = 0.
Orthogonal = Perpendicular = Normal
Ex. Show that 2i + 2j – k and 5i – 4j + 2k
are orthogonal
The direction angles of a vector a are the
angles α, β, and γ that a makes with the
positive x-, y-, and z-axes, respectively.
Consider α…
It’s the angle with the x-axis, so it’s the angle
with i.
a1
cos  
a
a3
cos  
a
a2
cos  
a
These are called the direction cosines of a.
a
 cos  ,cos  ,cos 
a
cos   cos   cos   1
2
2
2
Ex. Find the direction angles of a  1, 2,3
The vector projection of b onto a, written
projab, is the “shadow” that is cast by b onto a
b
proja b
a
The scalar projection (also called the
component) is the magnitude of the projection
Let’s find compab
b
proja b
a
a b
comp a b 
a
a b
a b
proja b  2 a or
a
aa
a
Ex. Find the scalar and vector projections of
b  1,1, 2 onto a  2,3,1