1. 12.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector onto another vector Work done by constant force

4. The angle between two nonzero vectors with the same initial point is the smallest angle between them.

5. Find the angle between the vectors v = and w =

6. Find the angle between the two vectors F 1 and F 2 where F 1 = j-k and F 2 = 2i-j+2k

8. Find a number k such that u = is orthogonal to v =

14. Find the direction cosines and angles of the vector v = (4, -2, -4)

16. Normal Vector (orthogonal to PQ)

18. Ex 2: Given a = and b =, find the vector projection of b onto a.

19. Which of the following does not make sense?

20. WORK: YOU HAVE TO CHANGE KINETIC ENERGY OF AN OBJECT TO DO WORK! IN OTHER WORDS: EXERTING FORCE DOES NOT EQUAL WORK! Physics

21. Application #1 A woman exerts a horizontal force of 25lb on a crate as she pushes it up a ramp that is 10 feet long and inclined at an angle of 20 degrees above the horizontal. Find the work done on the box.

22. Applications in real life #2: work W = (magnitude of force) (displacement) = |F||D|cos(theta) A mass is dragged up an incline of 38 degrees for 2 m by a force of 5.8 N that is directed at an angle of 54 degrees to the horizontal as shown in the diagram. What is the work done?

23. 12.3 HW Homework/Classwork