2. The definition of the product of two vectors is: This is called the dot product. Notice the answer is just a number NOT a vector.

3. What is the dot product of: and A. B. C.-11 D.5

4. What is the dot product of: and A.14 B.24 C.38 D.46

5. Properties of the Dot Product Let u, v, and w be vectors in the plane or in space and let c be a scalar.

6. The dot product is useful for several things. One of the important uses is in a formula for finding the angle between two vectors that have the same initial point. u v 

8. What is the angle between the vectors u = and v = ? A.60° B.14° C.40° D.17°

9. What is the angle between the vectors u = and v = ? A.78° B.100° C.82.9° D.93°

10. Find the angle between the vectors v = and w = The vectors have the same direction. We say they are parallel because remember vectors can be moved around as long as you don't change magnitude or direction. What does it mean when the angle between the vectors is 0?

11. Orthogonal (Perpendicular) Vectors Two vectors are orthogonal if their dot product is 0 Example:

12. Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal. The vectors v and w are orthogonal. If the angle between 2 vectors is, what would their dot product be? Since cos is 0, the dot product must be 0. Vectors u and v in this case are called orthogonal. (similar to perpendicular but refers to vectors). compute their dot product and see if it is 0 w = 2i + 8j v = 4i - j

13. Are the vectors and orthogonal? A.Yes B.No

14. Are the vectors and orthogonal? A.Yes B.No

15. The work W done by a constant force F in moving an object from A to B is defined as A use of the dot product is found in the formula below: This means the force is in some direction given by the vector F but the line of motion of the object is along a vector from A to B

16. Work Example Constant force of 40 pounds in the direction of 25 degrees with the horizontal. The object is moved 20 feet, what is the work done? Steps: Find the x component of the force: 40cos(25) = 36.25 Multiply by the distance: 20 feet 36.35(20)=725

17. Word Problem #1 An airplane with an airspeed of 200 mi/hr is flying N50E and a 40 mi/hr wind is blowing directly from the West. What is the ground speed of the plane and the true course.

18. Word Problem #2 An airplane flying in the direction S40E with an airspeed of 500 mi/hr and a 30 mi/hr wind is blowing in the direction N65E. Approximate the ground speed and the true course.

19. Word Problem #3 An airplane pilot wishes to maintain a true course in the direction S70W with a ground speed of 400 mi/hr when the wind is blowing directly north at 50 mi/hr. Approximate the required airspeed and heading.

20. Word Problem #4 An airplane is flying in the direction N20E with an airspeed of 300 mi/hr. Its ground speed and true course are 350 mi/hr and N30E, respectively. Approximate the direction and speed of the wind.

21. Word Problem #6 A quarterback releases a football with a speed of 50 ft/sec at an angle of 35 degrees with the horizontal. What are the horizontal and vertical components of the vector?

22. Word Problem #7 A child pulls a sled through the snow by exerting a force of 20 pounds at an angle of 40 degrees with the horizontal. What are the horizontal and vertical components of the vector?