Vectors

1. Vectors Day 1: Read pp. 45-53 and 62-64 in textbook Day 2: Vector Sheet A Day 3: Vector Sheet B

2. Warm Up.You may use your notes. 1. Draw a unique right triangle. Measure two sides. Solve the third side with Pythagorean equation. Show how you used SOH-CAH-TOA to calculate the angles. Verify that all angles sum to 180 degrees. 2. Draw a unique non-right triangle. Measure all three sides. Solve the interior angles. Show all your work. Label the Law of Cosines. Label the Law of Sines. Prove that the angles sum to 180 degrees. Grading: 10 points Individual. 10 points Cooperative Table Grade.

3. Introduction to Vectors {2010}: • Introduction: In physics, some physical quantities measure only an __________. Other physical quantities must also include a ______________ with the measured amount. The quantities that measure only an amount are called ___________. Those quantities that include a direction are called ____________. amount direction scalars vectors

4. Examples of scalars: mass, time, distance, speed, energy, temperature Examples of vectors: displacement, velocity, acceleration, force, momentum, fields (gravitional, electric and magnetic)

5. II. Representing Vectors: When representing a vector in mathematical equations, some kind of notation must be used to distinguish the vector from the scalar. Scalars will be represented with ordinary letters, whereas vectors will be represented by a letter with an arrow over the top of it. For example: = scalars = vectors When drawing a picture detailing a given problem, vectors are represented by arrows. The length of the arrow represents the amount of the vector, and the direction of the arrow gives the direction of the vector.

6. velocity of an airplane

7. sum of two displacements.

8. boat traveling across a stream, flow of river carries boat downstream. Note that the boat is not pointing in the direction it is traveling! We will see an example of this later…

9. III. Equality of Vectors: If two vectors have the same length and the same direction, then these two vectors are considered ___________. Any vector can be moved from one place to another, and so long as the magnitude (length) and direction are not changed, it will still be considered the ____________ vector. equal same

10. Even though each of these vectors start and finish in a different location, each has the same length and direction as the others. These vectors are all considered equal to one another. The picture can also be interpreted as the same vector drawn in 4 different locations.

11. Even though each of these vectors start and finish in a different location, each has the same length and direction as the others. These vectors are all considered equal to one another.

12. IV. Scalar Multiplication of Vectors: A vector may be multiplied by a scalar. Multiplying a vector by a positive scalar will only change the length (and possibly the unit) of the vector quantity, but it will not change the direction of the vector. Example #1: Given vector measures 3.00 cm towards the right, what is ?

13. IV. Scalar Multiplication of Vectors: A vector may be multiplied by a scalar. Multiplying a vector by a positive scalar will only change the length (and possibly the unit) of the vector quantity, but it will not change the direction of the vector. Example #1: Given vector measures 3.00 cm towards the right, what is ? is a vector 3 times as long as vector , in the same direction.

14. Just as a little extra, here is an example of a scalar multiplication that changes units: momentum is mass times velocity Since mass is a positive quantity, the momentum and the velocity vector always point in the same direction.

15. IV. Scalar Multiplication of Vectors: {continued…} A vector may also be multiplied by a negative scalar. The negative sign will reverse the direction of the vector. The size of the scalar will change the length of the vector. Example #2: Given vector measures 3.00 cm towards the right, what are and ?

16. The negative of a vector is a vector with the same length that points in the opposite direction.

17. V. Adding and Subtracting Vectors: Two vectors may be added together provided that the values of the vectors have the same units. The diagrams below show the addition of two vectors, and . The addition of two vectors can be viewed as a series of operations: first do vector #1 then do vector #2.

18. The tip – to – tail method of adding vectors places the tail (starting point) of one vector at the tip (ending point) of another vector. The resultant vector extends from the start of the first vector to the end of the second vector.

19. The parallelogram method of adding vectors places the tails (starting points) of both vectors at the origin. Recall that a parallelogram has two sets of parallel sides, with each member of the same set having the same length. Next place copies of each vector at the tip of the other vector to finish off the parallelogram. The resultant vector will extend from the origin out as a body diagonal. Note that when this is compared to the tip – to – tail method, the order of addition does not make any difference.

20. Why is this diagram incorrect? Vector ‘C’ does not represent the sum of the two vectors. This vector is the wrong body diagonal on the parallelogram method. This is a common mistake, though. Vector ‘C’ actually represents a difference, or subtraction, between the two vectors. From the tip – to – tail method, get:

21. This diagram shows again that the order of addition of vectors is not important. Note the parallelogram method for vector addition.

22. This diagram also shows that the order of addition of vectors is not important. Here, three vectors are added in various orders. The result is the same each time.

23. This diagram shows tip – to – tail method for adding three vectors.

24. Another diagram detailing order of addition…

25. This diagram details how to subtract two vectors. Just view the subtracted vector as adding the negative of the vector.

26. Another diagram showing how to interpret the subtraction of two vectors…

27. VI. Reporting Directions:Map Coordinates, Bearing, Azimuth Say a vector is given pointing as shown. How do we write the angle for this vector to show direction? N W E The first system will write the numeric value of the angle to a one side of a given axis. This example would be written as: 30 degrees to the north side of the east axis. S

28. Vector Directions: Map Coordinates N This angle is measured as EN for East of the North axis WN This angle is measured as NE for North of the East axis This angle is measured as NW for North of the West axis W E SE SW WS ES S

29. Vector Directions: Map Bearings

31. Vector Directions: Map Bearings N W E The angle is measured clockwise from the North axis. North is 0°, east is 90°, south is 180°, and west is 270°. S

32. N Vector Directions: Azimuth The angle is measured as degrees East or West of the North/South axis. 35° A The angle is written in the form N35°E, or S27°W. W E Vector A is measured as N35°E, or 35° East of the North axis. 27° Vector B is measured as S27°W, or 27° West of the South axis. S B

33. EXAMPLES: 3. Convert 17° SW in to a WS direction , a bearing, and an azimuth. N W E 17° SW This vector also measures as 180° + 73° = 253°as a bearing. 90° – 17° = 73° WS This vector also measures as S73°W as an azimuth. S

34. 4. Convert 34° ES in to a SE direction , a bearing, and an azimuth N Bearing = 90° + 56° = 146° E W 90° – 34° = 56° SE 34° ES Azimuth = S34° E S

35. 5. Convert a bearing of 52° in to two different directions and an azimuth N Bearing 52° is also 52° EN and azimuth N52°E Finally 90° – 52° = 38° NE W E S

36. 6. Convert a bearing of 306° in to two different directions and an azimuth N Bearing = 306° Direction = 306° – 270° = 36° NW Direction = 360° – 306° = 54° WN W E Azimuth = N54° W S

37. Angles measured from vertical and horizontal……. Vertical From vertical Above horizontal Horizontal Below horizontal

38. Vectors, Day #2 Tonight: Vector Sheet A Day 3: Vector Sheet B

39. VII: Combining vectors ____________________ : the combination of 2 or more vectors. (also called... _______, _________, ______) Resultant sum total net Example #6: A swimmer averages 3.00 km/h in still water. A river flows due east at a rate of 2.00 km/hr. Calculate the velocity of the swimmer if she swims East: Swimmer River Resultant is 3.00 km/hr + 2.00 km/hr = 5.00 km/hr East

40. Example #7: A swimmer averages 3.00 km/h in still water. A river flows due east at a rate of 2.00 km/hr. Calculate the velocity of the swimmer if she swims West: River Swimmer Resultant is 3.00 km/hr West + 2.00 km/hr East = 1.00 km/hr West Alternative: Let west be negative and east positive. The sum becomes: (-3.00 km/hr) + (+2.00 km/hr) = -1.00 km/hr = 1.00 km/hr West

41. Example #8: A swimmer averages 3.00 km/h in still water. A river flows due east at a rate of 2.00 km/hr. Calculate the velocity of the swimmer if she swims North. Calculate the answer both using (a) trig and (b) graphically. (b) Graphical Solution: Solve the vector sum by drawing the vectors on a piece of paper. Use a ruler and protractor to put a scale to the length and to make the angle measurements for the vectors. On your picture, draw the resultant vector and measure its length with the ruler and its direction with the protractor. Finally, convert that length with the scaling factor you used to draw the original vectors.

42. (a) Trig Solution: Draw the vectors Swimmer 3.00 km/hr North Draw the resultant By Pythagorean Theorem the resultant is: q River 2.00 km/hr East The angle is needed for direction:

43. Ex. #9: A swimmer can swim at a speed of 3.00 km/h relative to still water. The swimmer wishes to cross the river from the south bank to the north bank, but the river flows eastward at a rate of 2.00 km/h. (a) Which direction should the swimmer aim so that she can reach the opposite bank directly? (b) How fast is she effectively moving across the river? (c) If the river is 255 meters wide, how long will it take her to cross the river? Do not write all this, just think logically about it: First, the 3.00 km/hr is the speed of the swimmer relative to the water, independent of whether the water is moving relative to land! If the water is moving relative to land, then the velocity of the swimmer adds as a vector to the velocity of the water to make the overall velocity of the swimmer relative to the ground. Always use logic to solve these problems. Where is the swimmer starting? Where does the swimmer want to end? What does the swimmer have to do to make this possible? For example, this swimmer wants to go due north. The river flows to the east. Which way should the swimmer point to arrive due north? Directly across or at some angle? Why?

44. Since the swimmer wants to go north, but the current will move the boat to the east, the swimmer must turn to an angle opposite to the current. In other words, the boat must aim west of north! Set up the picture as follows: Velocity of water = 2.00 km/hr Direction of swimmer: Desired path of swimmer. Direction the swimmer must aim, also the speed of the swimmer relative to water. 3.00 km/hr This is not the actual path of the swimmer! q

45. As for the time to cross, this depends on how far across and how fast the swimmer actually crosses the river. Remember, the 3.00 km/hr is how fast the swimmer travels relative to the water, not the shore! Solve for the speed of the swimmer across the river with Pythagorean’s Theorem. 2.00 km/hr A = actual speed. 3.00 km/hr minutes

46. REVIEW PROBLEM: A boat that goes 8.00 km/h in still water is to cross a river 385 m wide. The river is flowing south with a velocity of 2.50 km/h. (a) If he starts from the west bank of the river, where should the river boat pilot aim the boat so as to go directly across the river? (b) How long will it take to cross the river? Boat v, where to aim River v q Desired v min

47. VECTORS DAY 2: Vectors not at right angles Vectors not at right angles: can be solved by making _________ and then applying ____________ (law of __________ and ____________). Example 1: A boy walks 1.55 km East and then 2.45 km 37.0° NE. Find his displacement (which includes both magnitude and direction!). triangle trigonometry sines cosines N Resultant 2.45 km 143.0° W 37.0° a 1.55 km E S

48. Solve for the resultant using the law of cosines: Solve for the direction angle, a, using the law of sines:

49. Example 2: Find the displacement: 122 km 37.0° NE, 175 km 17.5° WS N Upper right corner: 90 - 37.0 - 17.5 = 35.5° 37.0° 35.5° A =122 km 17.5° 37.0° W E a B=175 km R S

50. First, solve for R with law of cosines. Now solve for a. Since this may be an ambiguous case, solve using the law of cosines.