1. Warm up
What is a vector? What is a force? How do we use vectors when analyzing forces?

2. Forces in two dimensions: Last chapter, we looked at forces in one dimension. Example: You push on a box with a force of 10N east and your friend pushes on the same box with a force of 5N east, what is Fnet? You push with 10N east and friend pushes 5N west, what is Fnet? Now we will look at forces in more than one direction. You push 10N east and friend pushes 10N north. What is Fnet? We use vector addition and triangles.
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Vectors

3. Vocabulary Review New vector components vector resolution Vectors
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4. Vectors in Two Dimensions
You can add vectors by placing them tip-to-tail.
When you move a vector, do not change its length or direction.
The resultant of the vector is drawn from the tail of the first vector to the tip of the second vector.
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Vectors

5. Vectors in Two Dimensions
If vectors A and B are perpendicular, use the Pythagorean theorem to find the magnitude of the resultant vector.
Pythagorean Theorem R B A
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Vectors

6. Example
If you walk 10meters east and then 10 meters north, what is your displacement?
If you push on a box with a force of 10N east and you friend pushes on a box with a force of 10N north, what is the net force?
If the vectors are not at right angle to each other, we CANNOT use Pythagorean theorem.
If we only know 1 side of the triangle we can not use Pythagorean.
We need to use Trigonometry.

7. Trigonometry
Notes

8. The Trigonometric Functions we will be looking at
SINE
COSINE
TANGENT

9. The Trigonometric Functions
SINE
COSINE
TANGENT

10. SINE
Pronounced “sign”

11. COSINE
Pronounced “co-sign”

12. Pronounced “tan-gent”

13. Represents an unknown angle
Greek Letter q
Pronounced “theta”
Represents an unknown angle

14. hypotenuse
hypotenuse
opposite
opposite
adjacent
adjacent

15. We need a way to remember all of these ratios…

16. Some
Old
Horse
Came A
Hoppin’
Through
Our
Alley

17. Sin
SOHCAHTOA
Opp
Hyp
Cos
Adj
Hyp
Tan
Opp
Adj

18. Finding sin, cos, and tan. (Just writing a ratio or decimal.)

19. 10.8 9 A 6 Find the sine, the cosine, and the tangent of angle A.
Give a fraction and decimal answer (round to 4 places).
10.8 9 A 6
Shrink yourself down and stand where the angle is.
Now, figure out your ratios.

20. 24.5 8.2 23.1 Find the sine, the cosine, and the tangent of angle A
Give a fraction and decimal answer (round to 4 decimal places).
8.2 A
23.1
Shrink yourself down and stand where the angle is.
Now, figure out your ratios.

21. 11/2/15 Hand in your article review
Hand in your Newton’s laws handouts
Take out your notebook so we can get reorganized.

22. Finding a side. (Figuring out which ratio to use and getting to use a trig button.)

23. Ex: 1. Figure out which ratio to use. Find x
Ex: 1 Figure out which ratio to use. Find x. Round to the nearest tenth.
20 m x
Shrink yourself down and stand where the angle is.
tan 55 20 =
Now, figure out which trig ratio you have and set up the problem.

24. Ex: 2 Find the missing side. Round to the nearest tenth.
80 ft x
3.1 72
tan = 80 ÷
3.1 =
Shrink yourself down and stand where the angle is.
Now, figure out which trig ratio you have and set up the problem.

25. Ex: 3 Find the missing side. Round to the nearest tenth.
Shrink yourself down and stand where the angle is.
Now, figure out which trig ratio you have and set up the problem.

26. Ex: 4 Find the missing side. Round to the nearest tenth.
20 ft x

27. Finding an angle. (Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons.)

28. Use inverse function to solve for angle
sinθ=opposite
hypotenuse
If:
opposite=15m
hypotenuse=30m
sinθ=15m= 0.5
30m
θ=sin-1
θ=0.5(sin-1)=30° θ
hypotenuse
adjacent
opposite

29. Ex. 1: Find . Round to four decimal places.
17.2 9
tan-1
17.2 9
Shrink yourself down and stand where the angle is.
Now, figure out which trig ratio you have and set up the problem.
Make sure you are in degree mode (not radians).

30. Ex. 2: Find . Round to three decimal places.7
2nd
cos 7 23 ) 23
Make sure you are in degree mode (not radians).

31. Ex. 3: Find . Round to three decimal places.
200
400
2nd
sin
200
400 )
Make sure you are in degree mode (not radians).

32. When we are trying to find a side we use sin, cos, or tan.
When we are trying to find an angle
we use sin-1, cos-1, or tan-1.

33. Vectors in Two Dimensions
If the angle between the vectors is not 90°, you can use the law of sines or the law of cosines.
Law of sines
Law of cosines
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Vectors

34. Vectors in Two Dimensions
KNOWN
UNKNOWN
A = 20.0 N
B = 7.0 N
R = ?
θ = 180.0° − 30.0° = 150.0°
θR = ?
Use with Example Problem 1.
Problem
Find the magnitude of the sum of two forces, one 20.0 N and the other 7.0 N, when the angle between them is 30.0°.
Response
SKETCH AND ANALYZE THE PROBLEM
Draw a vector diagram and add the vectors graphically.
List the knowns and unknowns.
Draw the resultant vector.
Click to continue.
Place vectors tip to tail.
Click to continue.
Draw the initial vectors.
Click to continue. ?
7.0 N
30.0°
20.0 N
θ = 150.0°
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Vectors

35. Vectors in Two Dimensions
KNOWN
UNKNOWN
A = 20.0 N
B = 7.0 N
R = ?
θ = 180.0° − 30.0° = 150.0°
θR = ?
Use with Example Problem 1.
Problem
Find the magnitude of the sum of two forces, one 20.0 N and the other 7.0 N, when the angle between them is 30.0°.
SOLVE FOR THE UNKNOWN
Use the law of cosines.
Response
SKETCH AND ANALYZE THE PROBLEM
Draw a vector diagram and add the vectors graphically.
List the knowns and unknowns.
EVALUATE THE ANSWER
This answer is consistent with this problem’s vector diagram, which shows that the resultant should indeed be slightly greater in magnitude than the 20.0-N force. ?
7.0 N
20.0 N
θ = 150.0°
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Vectors

36. Vector Components Concepts in Motion Vectors
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Vectors

37. Vector Components
A vector can be broken into its components, which are a vector parallel to the x-axis and another parallel to the y-axis.
The process of breaking a vector into its components is sometimes called a vector resolution.
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Vectors

38. Vector Components
Complete the diagram using the terms positive and negative.
Click on a question mark to reveal the answer. ? ? ? ? ? ? ? ? ? ? ? ?
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Vectors

39. Algebraic Addition of Vectors
Two or more vectors (A, B, C, etc.) may be added by first resolving each vector into its x- and y-components. +y +x Cy C B By A Ay Ax Bx Cx
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Vectors

40. Algebraic Addition of Vectors
Two or more vectors (A, B, C, etc.) may be added by first resolving each vector into its x- and y-components.
The x- and y-components are added to form the x- and y-components of the resultant: +y +x By Ay Cy C B Ry
Rx = Ax + Bx + Cx A
Ry = Ay + By + Cy Bx Ax Cx Rx
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Vectors

41. Algebraic Addition of Vectors
Use the Pythagorean theorem to find the magnitude of the resultant vector: R2 = Rx2 + Ry2.
Use trigonometry to find the angle of the resultant vector: +y +x Ry A B C R Rx
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Vectors

42. Algebraic Addition of Vectors+x
(East) +y
(North)
Algebraic Addition of Vectors
θB = 135°
B = 7.3 m
Use with Example Problem 2.
Problem
Add the following two vectors via the component method:
A is 4.0 m south B is 7.3 m northwest By
Response
SKETCH AND ANALYZE THE PROBLEM
Establish a coordinate system.
Draw a vector diagram. Include the knowns and unknowns in your diagram.
Sketch the x-components and y-components of A and B. Bx
θA = 270°
A = 4.0 m Ay
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Vectors

43. Algebraic Addition of Vectors+x
(East) +y
(North)
Algebraic Addition of Vectors
B = 7.3 m
Use with Example Problem 2.
Problem
Add the following two vectors via the component method:
A is 4.0 m south B is 7.3 m northwest By
Response
SKETCH AND ANALYZE THE PROBLEM
Use your vector diagram as needed. Bx
Rx = −5.16 m
SOLVE FOR THE UNKNOWN
Find the x-components of A and B and add them to find the x-component of R.
Draw and label Rx on the vector diagram.
A = 4.0 m Ay
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Vectors

44. Algebraic Addition of Vectors+x
(East) +y
(North)
Algebraic Addition of Vectors
B = 7.3 m
Use with Example Problem 2.
Problem
Add the following two vectors via the component method:
A is 4.0 m south B is 7.3 m northwest By
Ry = 1.16 m
Response
SKETCH AND ANALYZE THE PROBLEM
Use your vector diagram as needed. Bx
Rx = −5.16 m
SOLVE FOR THE UNKNOWN
Find the y-components of A and B and add them to find the y-component of R.
Draw and label Ry on the vector diagram.
A = 4.0 m Ay
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Vectors

45. Algebraic Addition of Vectors+x
(East) +y
(North)
Algebraic Addition of Vectors
B = 7.3 m
Use with Example Problem 2.
Problem
Add the following two vectors via the component method:
A is 4.0 m south B is 7.3 m northwest
R = 5.3 m
Response
SKETCH AND ANALYZE THE PROBLEM
Use your vector diagram as needed.
θR = 167°
SOLVE FOR THE UNKNOWN
R = 5.3 m at 13° north of west
A = 4.0 m
EVALUATE THE ANSWER
Check the answer graphically.
Vectors
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46. Review Essential Questions Vocabulary
How are vectors added graphically?
What are the components of a vector?
How are vectors added algebraically?
Vocabulary
components
vector resolution
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Vectors