PHY1-11_12-Q1-0206-PF-FD.pptxhfgfdfgdfgg

Physics

1. General Physics 1 Science, Technology, Engineering, and Mathematics Lesson 2.6 Operations Using Unit Vectors

2. ‹#› Vectors are essential in the development of games, specifically 2D games. Vectors are applied in games to indicate the motion of an object.

3. ‹#› Numbers are used to indicate the thrust and the gravity that affects an object in situations where jumping is required.

4. ‹#› Vectors can also be used to specify the movement of an object from its original position.

5. ‹#› These vectors are incorporated in the code as numbers and can be used to manipulate different elements in the game.

6. ‹#› What are unit vectors? How can these unit vectors be used in mathematical operations?

7. ‹#› Calculate directions and magnitudes of vector (STEM_GP12V-Ia-11).

8. ‹#› Define a unit vector. Rewrite a vector using its components multiplied by unit vectors. Add and subtract vectors using the vector components. Calculate scalar and vector products using the vector components.

9. ‹#› Adding Vectors Graphically There are instances where the separation of magnitude and direction of vectors is convenient to use for basic calculations. This can be done using unit vectors. Unit vectors somehow “normalize” the vector such that the direction is retained but it can be easily scaled up or down by multiplying a scalar value to it.

10. ‹#› What is a unit vector?

11. ‹#› Unit Vector A unit vector is a vector that has a magnitude of 1 and has no units. Its main purpose is to specify the direction of a vector. A caret or “hat” (^) is placed above a boldface letter. It is used to differentiate vectors that may or may not have a magnitude of 1. Unit Vectors

12. ‹#› Finding the Unit Vector Any nonzero vector has an equivalent unit vector. It has the same direction as the vector but has a magnitude of 1. Unit Vectors

13. ‹#› Finding the Unit Vector Consider vector . You can solve the corresponding unit vector by using the expression below. Unit Vectors

14. ‹#› Writing the Unit Vector Components inside the brackets specify the position of the vector in the coordinate system. Unit Vectors vector standard position

15. ‹#› Writing the Unit Vector This vector is in standard position , which means that it started at the origin (0, 0). Unit Vectors vector standard position

16. ‹#› Writing the Unit Vector Unit Vectors vector in standard position

17. ‹#› The magnitude of vectors can be solved using the Pythagorean Theorem . If , then . Unit Vectors

18. ‹#› Then, . The unit vector can be expressed then as . Unit Vectors

19. ‹#› How do we check if the unit vector is indeed a unit vector? The magnitude of a unit vector is always one. Unit Vectors

20. ‹#› Vectors can also be expressed in terms of their unit vectors. Unit Vectors + x -direction + y -direction + z -direction

21. ‹#› How are mathematical operations performed in unit vectors?

22. ‹#› Vector Addition and Subtraction Consider two vectors given below. Operations involving Unit Vectors

23. ‹#› Vector Addition and Subtraction The resultant vector can be determined by adding components separately. Operations involving Unit Vectors R x R y

24. ‹#› Vector Addition and Subtraction The magnitude of the vector can be determined using the Pythagorean Theorem. Operations involving Unit Vectors

25. ‹#› Multiplying a Vector with a Scalar The scalar number can be multiplied to each of the components of the vector. Operations involving Unit Vectors

26. ‹#› Multiplying a Vector with a Scalar How do you compare the two vectors? How about these two vectors? Operations involving Unit Vectors

27. ‹#› Always remember the trigonometric functions (sine, cosine, and tangent) and the Pythagorean theorem as you proceed with the lesson. These concepts are essential in all the worked examples to follow.

28. ‹#› What is the unit vector of the vectors provided below?

29. ‹#› What is the unit vector of the vector provided below?

30. ‹#› ‹#› What is the unit vector of A = < 2, –3, 1 >?

31. ‹#› Three displacement vectors magnitudes A = 5, B = 10, and C = 20, respectively. Their directions are measured from the +x -axis with angles α = 45°, β = 200°, and 𝛾 = 30°, for vectors A, B, and C, respectively. (a) Find the resultant vector R = A + B + C in terms of unit vectors. (b) Find the magnitude and direction of the resultant vector R.

32. ‹#› The resultant vector is . The magnitude and direction of the resultant vector is 15.29, 41.44°. Three displacement vectors magnitudes A = 5, B = 10, and C = 20, respectively. Their directions are measured from the +x -axis with angles α = 45°, β = 200°, and 𝛾 = 30°, for vectors A, B, and C, respectively. (a) Find the resultant vector R = A + B + C in terms of unit vectors. (b) Find the magnitude and direction of the resultant vector R.

33. ‹#› ‹#› Find vector E = < 2, -1, 3 > using the given magnitudes and directions of vectors from Example 2. Specify the magnitude and direction of vector .

34. ‹#› Find the magnitude of vector that will satisfy: where and .

35. ‹#› Find the magnitude of vector that will satisfy: where and . The magnitude of the vector is equal to 7.07.

36. ‹#› ‹#› What is the magnitude of B to satisfy the equation 2A - B + 3C = < 4 >? Consider A = < 5, -2 > and C = < 3, 1, -3 >.

37. ‹#› Scalar Product using Components Consider the scalar product of two vectors. How do you multiply the terms of these vectors? Operations involving Unit Vectors

38. ‹#› Can you fill up the rest? The first one is provided as an example. Operations involving Unit Vectors

39. ‹#› The complete scalar product is given below. Operations involving Unit Vectors

40. ‹#› But recall that . If a unit vector is multiplied by itself, . Operations involving Unit Vectors

41. ‹#› But recall that . If a unit vector is multiplied by another unit vector, . Operations involving Unit Vectors

42. ‹#› Therefore, this equation can be reduced. Operations involving Unit Vectors

43. ‹#› Therefore, this equation can be reduced. Operations involving Unit Vectors

44. ‹#› How can you calculate the scalar product of two vectors?

45. ‹#› Find the scalar product of the vectors below: Both vectors are measured from the +x -axis.

46. ‹#› Find the scalar product of the vectors below: Both vectors are measured from the +x -axis. The scalar product of is 13.894.

47. ‹#› ‹#› What is the scalar product if vector A has a value 20 m and found at 100° and vector B has a value 30 m at 150°? Both angles are measured from the +x -axis.

48. ‹#› Find the scalar product if and . Find also the magnitudes of both each vector.

49. ‹#› Find the scalar product if and . Find also the magnitudes of both each vector. The scalar product is 9.00. The magnitudes of vectors A and B are 4.123 and 5.477, respectively.

50. ‹#› ‹#› What are the magnitudes of the following vectors: A = < 10, -20, -5 > B = < 30, 25, 10 > C = < 5, -10, 15 > What is the scalar product of vectors A and C?

51. ‹#› Find the angle 𝜙 between vectors: for .

52. ‹#› Find the angle 𝜙 between vectors: for . The angle 𝜙 between the two vectors is 148.38°.

53. ‹#› ‹#› What is the angle 𝜙 between the two vectors given below? C = < 20, -30, 10 > D = < 8, -10, -20 >

54. ‹#› Vector Product using Components The vector product using components can be achieved similarly to how scalar products are performed. But they follow different rules . Operations involving Unit Vectors

55. ‹#› R ecall that . If a unit vector is multiplied by itself, . Operations involving Unit Vectors

56. ‹#› Recall that . If a unit vector is multiplied by another unit vector, . Operations involving Unit Vectors

57. ‹#› The direction, on the other hand, can be determined using a right-handed system. Operations involving Unit Vectors

58. ‹#› Therefore, this equation can be reduced. Operations involving Unit Vectors

59. ‹#› Therefore, this equation can be reduced. Operations involving Unit Vectors C x C y C z

60. ‹#› Always keep in mind that the distributive property applies to both scalar and vector products. However, commutative property does not apply to vector product. This means that the order of multiplication matters. It may affect your final answer.

61. ‹#› Vector A has a magnitude of 20 and lies along the + x -axis. Vector B has a magnitude of 10, lies in the xy -plane, and makes 45° with the + x -axis. What is the vector product A x B?

62. ‹#› The vector product is < 141.42 >. Vector A has a magnitude of 20 and lies along the + x -axis. Vector B has a magnitude of 10, lies in the xy -plane, and makes 45° with the + x -axis. What is the vector product A x B?

63. ‹#› ‹#› What is the vector product A x B if vector A lies in the xy -plane, has a magnitude of 35, and makes an angle of 20° from the +x -axis, while vector while vector B lies along the +y -axis and has a magnitude of 55?

64. ‹#› Find the vector product A x B if A = < 1, 2, -3 > and B = < -5, 4, 1 >.

65. ‹#› The vector product is C = < 14, 14, 14 >. Find the vector product A x B if A = < 1, 2, -3 > and B = < -5, 4, 1 >.

66. ‹#› ‹#› What is the vector product C x B if B = < 15, -30, 10 > and C = < -10, 15, -25 >?

67. ‹#› Given vectors A = < 10, -6, -8 > and B = < -4, -9, 10 >, find (a) vector product A x B, and (b) the angle between vectors A and B.

68. ‹#› The vector product A x B = < -132, -68, -114 >. The angle 𝜙 between the two vectors is 70.58°. Given vectors A = < 10, -6, -8 > and B = < -4, -9, 10 >, find (a) vector product A x B, and (b) the angle between vectors A and B.

69. ‹#› ‹#› Three vectors A = < -5, -2, 0 >, B = < 2, 1, 6 >, and C = < 10, 0, -5 > are given. Find B x C, the angle between vectors B and C, and the angle between B x C and A.

70. ‹#› Fill in the missing word(s) to complete each statement. A unit vector has a magnitude of __________. A unit vector specifies the __________ of the vector. The magnitude of vectors can be determined using the _____________________.

71. ‹#› Solve the following problems. What is the equivalent unit vector of ? Find the equivalent unit vector of . Find the magnitude of the resultant if and .

72. ‹#› A unit vector is a vector that has no units but has a magnitude of 1. Its main purpose is to specify the direction of a vector. A caret or “hat” (^) is placed above a boldface letter. Vectors can be added and subtracted if their components are given in terms of unit vectors.

73. ‹#› The scalar product of two vectors is the sum of the products of their components. The vector product of two vectors can be determined by calculating the scalar components of the cross product vectors. The direction can be specified using the right-handed system.

74. ‹#› Concept Formula Description Operations Using Unit Vectors where is the unit vector is the vector is the magnitude of the unit vector Use this formula to determine the equivalent unit vector of a given vector.

75. ‹#› Concept Formula Description Operations Using Unit Vectors where is the vector is i th component of the vector, where i can be x, y, and z Use this formula to write the x -, y -, and z -components of a vector in terms of unit vectors.

76. ‹#› Concept Formula Description Operations Using Unit Vectors where and are vectors and are i th components of the vector, where i can be x, y, and z Use this formula to calculate the scalar product of two vectors.

77. ‹#› Concept Formula Description Operations Using Unit Vectors where and are vectors and are i th components of the vector, where i can be x, y, and z Use this formula to calculate the components of the magnitude of the vector product of two vectors.

78. ‹#› ‹#› Consider a nonzero vector D . How will you write its equivalent unit vector? How about if vector D has an angle 𝜃 with respect to the x -axis, what would be its direction?

79. ‹#› Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed) . Singapore: Brooks/Cole, 2006. Giancoli, Douglas C. Physics Principles with Applications (7th ed). USA: Pearson Education, 2014. Halliday, David, Robert Resnick and Kenneth Krane. Fundamentals of Physics (5th ed) . USA: Wiley, 2002. Knight, Randall D. Physics for Scientists and Engineers: A Strategic Approach (4th ed) . USA: Pearson Education, 2017. Serway, Raymond A. and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics (9th ed) . USA: Brooks/Cole, 2014. Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky’s University Physics with Modern Physics (13th ed) . USA: Pearson Education, 2012.