Chapter 6

1. Chapter 6 6-4 Vectors and dot products

2. Objectives • Find the dot product of two vectors and use the properties • Find the angle between two vectors • Write vectors as the sum of two vectors components • Use vectors to find the work done by a force

3. Dot product of two vectors • Dot Product is a third vector operation. This vector operation yields a scalar (a single number) not another vector. The dot product can be positive, zero or negative. • Definition of dot product

4. Example This is called the dot product. Notice the answer is just a number NOT a vector.

5. Example #1: Finding dot products Find each dot product. A) <4, 5>●<2, 3> sol: 23 B) <2, -1>●<1, 2> sol:0 C) <0, 3>●<4, -2> sol:-6 D) <6, 3>●<2, -4> sol: 0 E) (5i + j)●(3i – j) sol:14

6. Properties of Dot products Let u, v, and w be vectors in the plane or in space and let c be a scalar. • u●v = v●u • 0●v = 0 • u●(v + w) = u●v + u●w • v●v = ||v||2 • c(u●v) = cu●v = u●cv

7. Example#2 Using properties of dot products • Let u=<-1,3>, v=<2,-4> and w=<1,-2>. Use vectors and their properties to find the indicated quantity • A. (u.v)w sol: <-14,28> • B.u.2v sol: -28 • C.||u|| sol:

8. Check it out! Let u = <1, 2>, v = <-2, 4> and w = <-1, -2>. Find the dot product. A) (u●v)w B) u●2v

9. Check it out ! • Given the vectors u = 8i + 8j and v = —10i + 11j find the following. • A. • B.

10. Angle between two vectors • Angle between two vectors (θ is the smallest non-negative angle between the two vectors)

11. Example Find the angle between

12. solution

13. solution

14. Check it out! • Find the angle between u = <4, 3> and v = <3, 5>. • Solution: 22.2 degrees

15. Definitions of orthogonal vectors The vectors u and v are orthogonal if u●v = 0. Orthogonal = Perpendicular = Meeting at 90°

16. Example: Orthogonal vectors Are the vectors u = <2, -3> and v = <6, 4> orthogonal?

17. Example • Determine if the pair of vectors is orthogonal

18. Definition of Vector Components Let u and v be nonzero vectors. u = w1 + w2 and w1· w2 = 0 Also, w1 is a scalar of v The vector w1 is the projection of u onto v, So w1 = proj v u w 2 = u – w 1

19. Decomposing of a Vector Using Vector Components • Find the projection of u into v. Then write u as the sum of two orthogonal vectors • Sol:

20. Solution

21. Work • The work W done by a constant force F in moving an object from A to B is defined as 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

22. Definition of work Work is force times distance. If Force is a constant and not at an angle If Force is at an angle

23. Example • To close a barn’s sliding door, a person pulls on a rope with a constant force of 50 lbs. at a constant angle of 60 degrees. Find the work done in moving the door 12 feet to its closed position. • Sol: 300 lbs

24. Example • Find the work done by a force of 50 kilograms acting in the direction 3i + j in moving an object 20 metres from (0, 0) to (20, 0).

25. solution • Let's find a unit vector in the direction 3i + j Our force vector is in this direction but has a magnitude of 50 so we'll multiply our unit vector by 50.

26. solution

27. Student guided practice • Do7,9,11,23,31,43 and 57 in your book page 440 and 441

28. Homework • Do 8,10,12,14,24,32,44 and 69 in your book page 440 and 441

29. Closure • Today we learned about vectors and work • Next class we are going to learn about trigonometric form of a complex number