1. 1 st Day Section 6.4

2. Definition of Dot Product The dot product of vector u and vector v is A dot product is always a scalar (real #). Why?

3. Example 1 Find the following. a.u ∙ v b.u ∙ (v + w)

4. The Angle Between Two Vectors

5. The angle between two nonzero vectors is the angle θ, 0   θ   or 0°  θ  180°, that is between their respective standard position vectors. u v θ

6. Example 2

7. This formula can be rewritten to produce an alternative form of the dot product. END OF 1 ST DAY Friday’s HW: pp. 467-468 (4-48 mult. of 4)

8. 2 nd Day

9. What is the measure of θ, the angle between u and v in radians and what is cos θ? θ =  and cos θ = -1.

10. What would the range be for cos θ if the two vectors formed an obtuse angle? θ

11. What is the value of cos θ if θ is a right angle? cos θ = 0 θ

12. What would the range before cos θ if the two vectors formed an acute angle? θ

13. What is the measure of θ, the angle between u and v and what is cos θ if the vectors go in the same direction? θ = 0  and cos θ = 1. v

14. Orthogonal Vectors What does this mean about the zero vector and any vector u? What is the measure of the angle between two orthogonal vectors? The measure of the angle between two orthogonal vectors is 90° 0 They are orthogonal.

15. Example 1 Yes, u and v are orthogonal.

16. Applications

17. Example 2 A truck with a gross weight of 36,000 pounds is parked on a hill inclined at 10°. Assume that the only force to overcome is the force of gravity. Find the force, to the nearest tenth of a pound, required to keep the truck from rolling down the hill.

18. 10°F F = the force of gravity = v v = the force to keep the truck from rolling down the hill Draw a vector perpendicular to v.

19. 10°

20. Work The work W done by a constant force F acting along the line of motion of an object is given by

21. If the constant force F is not directed along the line of motion, the work W done by the force is given by

22. Example 3 A man pushes a broom with a constant force of 40 pounds. The handle of the broom is at an angle of 30°. How much work, to the nearest foot-pound, is done pushing the broom 30 feet? pp. 468-469 (60-74 even)