1. 6-4 Vectors and dot products
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:√10

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