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Vectors Everything you need to know. Topic 1.3 Vector Basics Scalar & Vector Quantities Magnitude - the size of the number Example: 1.60x10 -19 ; 55.

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Presentation on theme: "Vectors Everything you need to know. Topic 1.3 Vector Basics Scalar & Vector Quantities Magnitude - the size of the number Example: 1.60x10 -19 ; 55."— Presentation transcript:

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2 Vectors Everything you need to know. Topic 1.3

3 Vector Basics Scalar & Vector Quantities Magnitude - the size of the number Example: 1.60x10 -19 ; 55 o ; 512 Scalar - a measurement with only magnitude Example: q = 1.60x10 -19 C; 55 o angle; 512 Hz

4 Vector - a measurement with both magnitude and direction Example: 25 mph, West; 10 m/s 2, downward; 10 N, right - vectors are drawn as an arrow - vectors are drawn as an arrow - vectors have a head and a tail - vectors have a head and a tail Scalar & Vector Quantities tail head tail head tail head tail head

5 - angles are measured counterclockwise from the horizontal. Length of the arrow shows the magnitude Direction shows the angle measurement 0o0o 90 o 180 o 270 o

6 40 o 0o0o 90 o 270 o 180 o 0.04m Vector is 0.04 m, 40 o A Page 1

7 240 o 0.05m Vector is 0.05 m, 240 o B 0o0o 90 o 270 o 180 o Page 1

8 161 o 0o0o 90 o 270 o 180 o 0.07m Vector is 0.07 m, 161 o C Page 1

9 60 o 7 km Vector is 7 km, 60 o D 0o0o 90 o 270 o 180 o Page 1

10 Standard Abbreviations page 2 Quantity UnitAbbreviations Distancemeterm Distance kilometerkm Speedmeters per secondm/s Speed kilometers per hourkm/h Acceleration meter per second squaredm/s 2 Force (weight)NewtonsN

11 0o0o 90 o 270 o 180 o 3 cm Vector is 3 cm, 90 o 90 o Page 3-4 A

12 0o0o 90 o 270 o 180 o 6 cm Vector is 6 cm, 150 o 150 o Page 3-4 D

13 0o0o 90 o 270 o 180 o 3.5 cm Vector is 3.5 cm, 215 o 215 o Page 3-4 L

14 0o0o 90 o 270 o 180 o 2 cm Vector is 2 cm, 310 o 310 o Page 3-4 Q

15 Algebraic Addition of Colinear Vectors (Page 7) 4 cm, 0 o + 5 cm, 0 o The sum or total is... From start to finish 9 cm, 0 o tail head tail head start finish

16 2 m, 90 o +5 m, 90 o 3 m, 90 o tail head tail head start finish

17 15 m/s 2, 180 o + 11 m/s 2, 0 o The sum or total is... From start to finish 4 m/s 2, 180 o The sum of two or more vectors is called the RESULTANT tail head tail head start finish

18 Class work Remember DO NOT WRITE IN PACKETS! Set A: Page 3 A-D Page 4 J-M Page 5 A-F Page 8 #1-5 Page 9 #11-15

19 SET A - answers Page 3 A. 3 cm, 90 o B. 4 cm, 0 o C. 2 cm, 180 o D. 6 cm, 150 o Page 4 J. 5.5 cm, 300 o K. 8 cm, 280 o L. 3.5 cm, 215 o M. 7.5 cm, 225 o

20 SET A - answers Page 5 A-F 0o0o 12 km A 36 o B 0o0o 24 km 115 o 0o0o 525 km C 213 o

21 SET A - answers Page 5 A-F (continued) 0o0o 900 km D 340 o E 0 o or East 30 km 0o0o 380 km F 270 o or South

22 SET A - answers Page 8 #1-5 1.) 3.0 m/s, 270 o 2.) 0.8 N, 0 o 3.) 1.2 m/s 2, 90 o 4.) 0.8 cm, 0 o 5.) 750 m, 0 o Page 9 #11-15 11.) 80 m, n 12.) 400 m/s, e 13.) 50 N, n 14.) 100 m, e 15.) 13 km, s

23 Pages 10-11 Practice with Calculators – Squaring and Square Roots We don’t need to do this... Right? Now... Vectors at right angle to each other are NOT colinear.

24 Adding of Vectors Using the Pythagorean Theorem (triangles with 90 o angles) 25 N, 0 o + 35 N, 90 o = _____?______ 1.) First draw the vectors head to tail. DO NOT PICK UP YOUR PENCIL! 2.) Then draw the resultant from start to finish. 0o0o 90 o 270 o 180 o 25 N, 0 o 35 N, 90 o Resultant start finish

25 Pythagorean Theorem a 2 + b 2 = c 2 c = √(a 2 + b 2 ) This is how you find the magnitude of the hypotenuse. A B C

26 25 N, 0 o + 35 N, 90 o = ____?_____ c = √ (a 2 + b 2 ) c = √ (25 2 + 35 2 ) c = √ 1850 c = 43 N The magnitude of the resultant is 43 N.

27 Names of the Sides of a Right Triangle The side next to the inside angle is called the adjacent side (Adj). The side across from the inside angle is called the opposite side (Opp). The longest side across from the right angle is called the hypotenuse (Hyp). Adjacent Opposite Hypotenuse 

28 Sine Using the opposite side and the hypotenuse you can find the inside angle. Sin  = Opp Hyp Adjacent Opposite Hypotenuse 

29 Cosine Using the adjacent side and the hypotenuse you can find the inside angle. Cos  = Adj Hyp Adjacent Opposite Hypotenuse 

30 Tangent Using the opposite side and the adjacent you can find the inside angle. Tan  = Opp Adj Adjacent Opposite Hypotenuse 

31 To help us remember this we use the mnemonic SOH CAH TOA Sin  = Opp Cos  = Adj Tan  = Opp Hyp Hyp Adj This is how you find the direction of angle measurement of the resultant.

32 25 N, 0 o + 35 N, 90 o = ____?_____ We already found the magnitude of the resultant to be 43 N. Now we have to find the direction of the resultant. We have the adjacent side 25 N and the opposite side 35 N so we will use the tangent to find the angle (i.e. direction). Adj Opp Hyp 

33 25 N, 0 o + 35 N, 90 o = ____?_____ Tan  = Opp. Adj.  = Tan -1 Opp. Adj.  = Tan -1 35 25  = 54 o F.Y.I. – All of our angles will be measured in degrees... Your calculator needs to be in degree mode. 0o0o 90 o 270 o 180 o 25 N, 0 o 35 N, 90 o Resultant start finish Adjacent Opposite

34 25 N, 0 o + 35 N, 90 o = ____?_____ The answer is... the sum of these two vectors, called the resultant, is 43 N, 54 o.

35 Quadrants Dou you remember quadrants? What happens if our vectors are not in the first quadrant? 0o0o 90 o 270 o 180 o I II III IV

36 180 km, 180 o + 90 km, 90 o = ?. DRAW FIRST!!! Find the magnitude. c = √(a 2 + b 2 ) c = √ (180 2 + 90 2 ) c = 201 km 0o0o 90 o 270 o 180 o 180 km, 180 o 90 km, 90 o Resultant

37 180 km, 180 o + 90 km, 90 o = ?. 0o0o 90 o 270 o 180 o 180 km, 180 o 90 km, 90 o Resultant Find the direction. Tan  = Opp. Adj.  = Tan -1 90 180  = 26.6 o The direction is 180 o – 26.6 o  = 153.4 o Adjacent Opposite  =26 o

38 180 km, 180 o + 90 km, 90 o = ?. The answer is... the sum of these two vectors, called the resultant, is 201 km, 153.4 o.

39 340 km, 0 o + 290 km, 270 o = ?. DRAW FIRST!!! Find the magnitude. c = √ (340 2 + 290 2 ) c = 447 km 0o0o 90 o 270 o 180 o 340 km, 0 o 290 km, 270 o Resultant

40 340 km, 0 o + 290 km, 270 o = ?. Find the direction. Tan  = Opp. Adj.  = Tan -1 290 340  = 40.0 o The direction is 360 o – 40.0 o  = 320 o Adjacent Opposite 0o0o 90 o 270 o 180 o 340 km, 0 o Resultant 290 km, 270 o  =40 o

41 340 km, 0 o + 290 km, 270 o = ?. The answer is... the sum of these two vectors, called the resultant, is 447 km, 320 o.

42 HOMEWORK Remember DO NOT WRITE IN PACKETS! Finish Set A Finish Set A and Set B: Page 20 ODD’s

43 SET B - answers Page 20 ODD’s 1.) 66.7 cm, 209.6 o 3.) 51.5 m/s, 209.1 o 5.) 101.7 m, 150 o 7.) 60.2 cm, 122.1 o 9.) 700.7 m, 44.5 o 11.) 127 km, 49.8 o 13.) 58.7 m, 113.1 o 15.) 65 km, 247.4 o

44 Skip pages 21-28 on the Law of Cosines and Law of Sines

45 Resolving Vectors into Components To Find the X-component: Cos  = Adj Hyp Solve for the adjacent. Adj = Hyp Cos  Adj = X-comp 0o0o 90 o 270 o 180 o X-component Y-component Vector Adjacent Opposite Hypotenuse 

46 Resolving Vectors into Components To Find the Y-component: Sin  = Opp Hyp Solve for the adjacent. Opp = Hyp Sin  Opp = Y-comp 0o0o 90 o 270 o 180 o X-component Y-component Vector Adjacent Opposite Hypotenuse 

47 760 N, 80 o To Find the X-component: Cos  = Adj Hyp Solve for the adjacent. Adj = Hyp Cos  Plug in the values. Adj = 760 Cos 80 o Adj = X-comp = 132 N 0o0o 90 o 270 o 180 o X-component Y-component 760 N, 80 o Adjacent Opposite Hypotenuse 

48 760 N, 80 o To Find the Y-component: Sin  = Opp Hyp Solve for the adjacent. Opp = Hyp Sin  Plug in the values. Opp = 760 Sin 80 o Opp = Y-comp = 748 N 0o0o 90 o 270 o 180 o X-component Y-component 760 N, 80 o Adjacent Opposite Hypotenuse 

49 760 N, 80 o The components of this vector are x-component is 132 N y-component is 748 N

50 To Find the X-component: Cos  = Adj Hyp Solve for the adjacent. Adj = Hyp Cos  Plug in the values. Adj = 8.8 Cos 210 o Adj = X-comp = -7.6 km 0o0o 90 o 270 o 180 o X-component Y-comp 8.8 km, 210 o Adjacent Opp Hypotenuse  8.8 km, 210 o

51 To Find the Y-component: Sin  = Opp Hyp Solve for the adjacent. Opp = Hyp Sin  Plug in the values. Opp = 8.8 Sin 210 o Opp = Y-comp = -4.4 km 0o0o 90 o 270 o 180 o X-component 8.8 km, 210 o Adjacent Hypotenuse  Y-comp Opp 8.8 km, 210 o

52 The components of this vector are x-component is -7.6 km y-component is -4.4 km

53 Class work Remember DO NOT WRITE IN PACKETS! Set C: Page 32ODD’s

54 SET C – answers SET C – answers Page 32 ODD’s 1.) x = 58.2 cm y = 48.9 cm 3.) x = 31.9 N y = 11.6 N 5.) x = 4.87 m/s y = 92.9 m/s 7.) x = -47.6 cm y = 56.7 cm 9.) x = -22.8 km y = 48.9 km 11.) x = -82.7 m/s y = -30.1 m/s 13.) x = -8.5 km y = -48.2 km 15.) x = -50.9 cm y = -50.9 cm 17.) x = 36.5 km y = -63.2 km 19.) x = 70 m/s 2 y = -25 m/s s

55 Page 34 Adding more than two vectors that are not colinear nor are they at right angles to each other. Adding Vectors Using the Resolution Method

56 54 N, 210 o + 54 N, 75 o + 92 N, 30 o = ? 0o0o 90 o 270 o 180 o We are trying to find the magnitude and direction of the resultant.

57 Find the x & y components of each vector 54 N, 210 o + 54 N, 75 o + 92 N, 30 o = ?. x-componentsy-components 54 cos210 o = -47 N54 sin210 o = -27 N 54 cos75 o = 14 N54 sin75 o = 52 N 92 cos30 o = 80 N92 sin30 o = 46 N 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o

58 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o Sum of the x-components Sum of the y-components Resultant

59 54 N, 210 o + 54 N, 75 o + 92 N, 30 o = ?. x-componentsy-components 54 cos210 o = -47 N54 sin210 o = -27 N 54 cos75 o = 14 N54 sin75 o = 52 N 92 cos30 o = 80 N92 sin30 o = 46 N +_____________ +_______________ 47 N71 N 0o0o 90 o 270 o 180 o Sum of the x-components Sum of the y-components Resultant 47 N 71 N c = √ (a 2 + b 2 ) c =√ (47 2 + 71 2 ) c = 85 N  = Tan -1 71 47  = 57 o

60 The resultant is 85 N, 57 o 0o0o 90 o 270 o 180 o 54 N, 210 o + 54 N, 75 o + 92 N, 30 o = ?

61 83 km, 160 o + 68 km, 240 o + 75 km, 50 o ? 0o0o 90 o 270 o 180 o

62 83 km, 160 o + 68 km, 240 o + 75 km, 50 o = ?. x-componentsy-components 83 cos160 o = -78 N83 sin160 o = 28 N 68 cos240 o = -34 N68 sin240 o = -59 N 75 cos50 o = 48 N75 sin50 o = 57 N 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o

63 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o Sum of the x-components Sum of the y-components Resultant

64 83 km, 160 o + 68 km, 240 o + 75 km, 50 o = ?. x-componentsy-components 83 cos160 o = -78 N83 sin160 o = 28 N 68 cos240 o = -34 N68 sin240 o = -59 N 75 cos50 o = 48 N75 sin50 o = 57 N +_____________ +_______________ -64 km26 km -64 km 26 km c = √ (a 2 + b 2 ) c =√ (-64 2 + 26 2 ) c = 69 km  = Tan -1 26 64  = 22 o 0o0o 90 o 270 o 180 o Sum of the x-components Sum of the y-comp Resultant  = 22 o  = 180 o – 22 o = 158 o

65 83 km, 160 o + 68 km, 240 o + 75 km, 50 o ? 0o0o 90 o 270 o 180 o The resultant is 69 km, 158 o

66 HOMEWORK Remember DO NOT WRITE IN PACKETS! Set D: Page 38#1-5

67 Page 38 #1 15 N, 80 o + 42 N, 45 o + 24 N, 20 o = ?. 15 N, 80 o + 42 N, 45 o + 24 N, 20 o ? 0o0o 90 o 270 o 180 o

68 15 N, 80 o + 42 N, 45 o + 24 N, 20 o = ?. x-componentsy-components 15 cos80 o = 2.6 N15 sin80 o = 14.8 N 42 cos45 o = 29.7 N42 sin45 o = 29.7 N 24 cos20 o = 22.6 N24 sin20 o = 8.2 N 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o 0o0o 90 o 270 o 180 o

69 0o0o 90 o 270 o 180 o Sum of the x-components Sum of the y-components Resultant

70 15 N, 80 o + 42 N, 45 o + 24 N, 20 o = ?. x-componentsy-components 15 cos80 o = 2.6 N15 sin80 o = 14.8 N 42 cos45 o = 29.7 N42 sin45 o = 29.7 N 24 cos20 o = 22.6 N24 sin20 o = 8.2 N +_____________ +_______________ 54.9 N52.7N 54.9 N 52.7N c = √ (a 2 + b 2 ) c =√ (54.9 2 + 52.7 2 ) c = 76.1 N  = Tan -1 52.7 54.9  = 43.8 o Sum of the x-components Sum of the y-comp 0o0o 90 o 270 o 180 o Resultant

71 Page 38 #1 15 N, 80 o + 42 N, 45 o + 24 N, 20 o = ?. 15 N, 80 o + 42 N, 45 o + 24 N, 20 o ? 0o0o 90 o 270 o 180 o The resultant is 76.1 N, 43.8 o

72 SET D - answers Page 38 #1-5 1.)76.1 N, 43.8 o 2.) 150.0 km, 62.9 o 3.) 145.2 m/s, 46.8 o 4.) 63.8 cm, 170.3 o 5.) 0.69 m/s 2, 157.9 o

73 Class work Remember DO NOT WRITE IN PACKETS! Set E: Page 38#6-15

74 SET E - answers Page 38 #6-15 6.)40.1 m, 205 o 7.) 52.9 N, 118 o 8.) 146 km, 111 o 9.) 17.2 m/s, 33.1 o 10.) 48.4 cm, 105 o 11.) 10.6 m/s 2, 25.6 o 12.) 38.5 m, 166 o 13.) 172 N, 47.3 o 14.) 47.2 km, 286 o 15.) 767 m/s, 355 o

75 The End

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