1. Vectors and Angles
Lesson 10.3b

2. Angle Between Two Vectors
Given vectors v and w
form angle 
We can show that 

3. Angle Between Two Vectors
Try these two vectors
What is the angle between?
1.47 radians or degrees

4. Orthogonal Vectors When the dot product equals zero …
What happens to the angle?
Orthogonal

5. Orthogonal Vectors
Are these two vectors orthogonal?

6. Projections Consider v and w, vectors in 2-space v w
u = projection of v on w
Note that u = t • u (for some scalar, t)

7. Projections We can say (v – t w) • w = 0 v – t • w (orthogonal to w) v
u = projection of v on w
We can say (v – t w) • w = 0

8. This is the scalar projection of v on w.
Projections
Since (v – t w) • w = 0
This is the scalar projection of v on w.
(This is a number.)

9. The end result is a vector.
Projections
The vector projection of v in the direction of w is
This is a scalar
The end result is a vector.

10. Work as a Dot Product Work = Force • Distance
But … what if force not in same direction of movement? F PQ

11. Work as a Dot Product Example: Wind = F Boat movement = PQ
Projection of F on PQ is the force used in direction of movement
Work = F • PQ the dot product F PQ

12. Work as a Dot Product Given F = 2i + 3j + 1k acting on a particle
Particle moves from P(1, 0, -1) to Q(3, 1, 2)
What is the work accomplished?

13. Assignment Lesson 10.3b Page 694
Exercises 11, 13, 31, 33, 35, 37, 39, 41, 45, 49, 51, 53, 55