12.3
The Dot Product
Definitions
Let u  u1 , u2 , v  v1 , v2 .
The dot product of u and v in the plane is
u  v  u1v1  u2 v2
(Read “u dot v”)
The dot product of u and v in space is
u  v  u1v1  u2 v2  u3 v3
Examples
3, 4  5, 2   3 5   4  2   23
2, 3  3, 2   23   3 2  0
Two vectors u and v are orthogonal
 if they meet at a right angle.
 if and only if u ∙ v = 0 (since slopes are opposite reciprocal)
Properties
Let u, v, w be vectors
1.
2.
3.
4.
u  v  v u
u  ( v  w)  u  v  u  w
c(u  v)  cu  v  u  cv
0 v  0
5. v  v  v
2
Another form of the Dot Product:
u  v  u v cos
where  is the angle between two nonzero vectors u and v.
Examples
Find the angle between vectors u and v:
u  2,3 , v  2,5
 uv 
11 
1 
  cos 
  cos 
 55.5

 13 29 
u v
1
Direction Cosines
Angles between a vector v and 3 unit vectors i, j and k
are called direction angles of v, denoted by α, β, and γ
respectively. Since
v  i  v cos   v1 , v2 , v3  1,0,0  v1
we obtain the following 3 direction cosines of v:
v1
cos  
v
v2
cos  
v
v3
cos  
v
Projections
𝑢∙𝑣
𝑝𝑟𝑜𝑗𝑣 𝒖 =
𝑣
2
𝑣
𝑢∙𝑣
𝑐𝑜𝑚𝑝𝑣 𝒖 =
𝑣
u
w
v
Let u and v be nonzero vectors.

w is called the vector projection of u onto v, denoted
by projvu

The signed magnitude of the vector projection is
called the scalar projection of u onto v, or vector component
of u along v, denoted by compvu
Examples
u  1,7, 2 , v  2, 2,6 , w  1, 1,3
1) Compute
u  ( v  w)
2) Compute
u v uw
3) List all pairs of orthogonal and/or parallel vectors.
4) Find the angle between vectors v and w.
5) Find the unit vector in the direction u.
6) Find scalar projection of w onto u.
7) Find vector projection of w onto u.