Dot Products

1
Dot Products

• Lecture 20
• Fri, Oct 12, 2007

2
The Dot Product

• Let u (u1, u2, u3) and v (v1, v2, v3).
• The dot product of u and v is defined to be
• u?? v u1v1 u2v2 u3v3.
• Note that the dot product is a real number.

3
Algebraic Properties of the Dot Product

• Let u, v, and w be vectors and let c be a real
  number and let ? be the angle between u and v.

4
Dot Products and Angles

• A consequence of the last property is that
• u ? v gt 0 if and only if 0? ? ? lt 90? (acute
  angle).
• u ? v 0 if and only if ? 90? (right angle).
• u ? v lt 0 if and only if 90? lt ? ? 180? (obtuse
  angle).
• This is of great importance in computer graphics.

5
Dot Products and Angles

• Let n be a vector that is normal (perpendicular)
  to a surface at a point, let L be a vector from
  that point to the light source, and let v be the
  vector from that point to the viewpoint.
• What is the significance of n ? L gt 0?
• What is the significance of n ? v gt 0?

6
Orthogonal Projections

• The orthogonal projection of u onto v is the
  vector
• Example Project u (5, 0, 2) onto v (3, 4,
  5).

7
Calculating Reflections

• Let L be the light vector from the surface to
  the light source.
• Let N be the unit normal vector from the surface.
• Let R be the reflection vector.

8
Calculating Reflections

• The vectors L and N.

N
L
surface
9
Calculating Reflections

• Let u be the projection of L onto N.

N
u
L
surface
10
Calculating Reflections

• Let v u L.

N
v
u
L
surface
11
Calculating Reflections

• Then the reflection vector is R u v.
• R 2(N?L)N L

N
v
v
u
R
L
surface
12
Viewing, Lighting, and Reflection Calculations

• Let the surface normal be (2/3, 2/3, 1/3) at the
  point P (5, 10, 10).
• Let the light vector be (0, 3/5, 4/5).
• Let the view point be V (-5, 5, 10).

13
Viewing, Lighting, and Reflection Calculations

• Is P visible from V?
• Is the light illuminating P?
• What is the reflection vector R?
• What is the angle between the view vector V and
  the reflection vector R?