1. VECTORS AND SCALARS UNIT II FUNDAMENTALS AND OPERATIONS

2. VECTORS AND DIRECTION In the study of motion we use quantities such as distance, displacement, speed, velocity, acceleration, etc. These can be divided into two categories: vector and scalar A vector is a quantity that is described by magnitude (number) and direction. A scalar is a quantity that is described by its magnitude.

3. VECTORS AND DIRECTION Examples of vectors include: displacement, velocity, acceleration, and force Vector quantities require both magnitude and direction: Examples: 20 meters and 30 degrees to the west of north. Vectors are often represented by scaled vector diagrams. Involves an arrow drawn to scale and in a specific direction. Commonly called free-body diagrams. Scale is clearly listed A vector arrow (with arrowhead) is drawn in a specified direction. The arrow has a head and a tail. The magnitude and direction of the vector are clearly labeled.

4. EXAMPLE OF VECTOR DIAGRAM Image from The Physics Classroom

5. Example: Due West, Due North, Due South, Due East Also can use Due North West, Due North East, Due South West, etc. The direction of a vector is often expressed as an angle of rotation of the vector about its “tail” from east, west, north, or south. Example: Direction of 40 degrees North of West which means a vector pointing West has been rotated 40 degrees towards the northerly direction. The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its “tail” from due East. Example: A vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east. HOW TO DESCRIBE THE DIRECTION OF VECTORS Image from The Physics Classroom

6. REPRESENTING MAGNITUDE The magnitude is depicted by the length of the arrow. Arrows are drawn according to the scale that is chosen. How long would the arrow be in the example given? What would the length of the arrow be if the displacement vector was 15 miles and the scale was 1 cm = 5 miles? Image from The Physics Classroom

7. RESULTANTS The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together. Example: If you add A, B, and C, the result will be vector R. When displacement vectors are added, the result is a resultant displacement. Any two vectors can be added as long as they are the same quantity. Example: Two or more velocity vectors are added, then you have a velocity resultant. “To do A + B + C is the same as to do R” Image from The Physics Classroom

8. VECTOR ADDITION Two vectors can be added together to determine the result or resultant. Examples below of how to add vectors: First vector + second vector = total If opposite direction, then subtract Image from The Physics Classroom

9. VECTOR ADDITION More complicated cases need special directions: The Pythagorean theorem and trigonometric methods The head to tail method using a scaled vector diagram Examples of more complicated cases below: Image from The Physics Classroom

10. THE PYTHAGOREAN THEOREM Only used when the result of adding two (and only two) vectors that make a right angle to each other. Cannot be used for adding more than two vectors or adding vectors that are not at 90 degree angles to each other. Image from The Physics Classroom

11. PYTHAGOREAN THEOREM EXAMPLE Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric’s resulting displacement. Image from The Physics Classroom

12. PYTHAGOREAN THEOREM EXAMPLE #2 In each example below, determine the magnitude of the vector sum. Image from The Physics Classroom

13. USING TRIGONOMETRY TO DETERMINE VECTOR’S DIRECTION The direction of a resultant vector can often be determined by use of trigonometric functions: SOH CAH TOA – Sine, Cosine, and Tangent The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The tangent function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

14. TRIGONOMETRIC FUNCTIONS Image from The Physics Classroom

15. EXAMPLE OF USING TRIG FUNCTIONS Apply trig functions to the hiker problem to determine the hiker’s overall displacement. Begin by selecting one of the two angles other than the right angle. Then use any of the three trig functions to find the measure of the angle. Once the measure of the angle is determined, the direction of the vector can be found. The vector makes an angle of 45 degrees with due East. (The direction of a vector is the counterclockwise angle of rotation that the vector makes with due East. Image from The Physics Classroom

16. EXAMPLE OF USING TRIG FUNCTIONS Sometimes the measure of an angle as determined by SOH CAH TOA is NOT always the direction of the vector. How would you determine direction? The following example illustrates this relationship. Image from The Physics Classroom

17. USE OF SCALED VECTOR DIAGRAMS TO DETERMINE A RESULTANT Must have accurately drawn scaled vector diagrams. Use the head to tail method. A common Physics lab involves a vector walk. Either using centimeter sized displacements on a map or meter-sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position. Example of the head to tail method after you have completed a vector walk. Image from The Physics Classroom

18. STEPS TO HEAD TO TAIL METHOD 1.Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper. 2.Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram. (SCALE: 1 cm = 20 m) 3.Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram. 4.Repeat steps 2 and 3 for all vectors that are to be added. 5.Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector the resultant or simply R. 6.Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (ex. 4.4 cm x 20 m/1cm = 88 m) 7.Measure the direction of the resultant using the counterclockwise convention.

19. HEAD TO TAIL EXAMPLE Calculate the resultant given the following vectors: Image from The Physics Classroom

20. HEAD TO TAIL EXAMPLE SOLUTION The order in which the three vectors are added has not affect upon either the magnitude or the direction of the resultant. Image from The Physics Classroom

21. VECTOR COMPONENTS Examples of vectors that are directed in two dimensions – upward and rightward or northward and eastward, etc. Use math to transform a vector into two parts along a coordinate axes. Example: A vector directed northwest has two parts – northward part and westward part. Example - Image from The Physics Classroom

22. VECTOR COMPONENTS Each part of a two dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equal to the influence of the single two dimensional vector. Image from The Physics Classroom

23. VECTOR COMPONENT EXAMPLE Consider a picture that is hung to a wall by means of two wires that are stretched vertically and horizontally. Each wire exerts a tension force on the picture to support its weight. Since the wire is stretched in two dimensions them it is a component vector. Image from The Physics Classroom

24. VECTOR COMPONENT EXAMPLE This means that the wire on the left could be replaced with two wires one pulling leftward and one pulling upward. Image from The Physics Classroom

25. PERPENDICULAR COMPONENTS The two perpendicular parts or components of a vector are independent of each other. A change in one component does not affect the other component. Changing a component will affect the motion in that specific direction. While the change in one of the components will alter the magnitude of the resulting force, it does not alter the magnitude of the other component. Example: an air balloon descending through the air toward the ground in the presence of a wind that blows eastward. Suppose that the downward velocity of the balloon is 3 m/s and that the wind is blowing east with a velocity of 4 m/s. The resulting velocity of the air balloon would be the combination (i.e., the vector sum) of these two simultaneous and independent velocity vectors. The air balloon would be moving downward and eastward.

26. PERPENDICULAR COMPONENTS If the wind velocity increased, the air balloon would begin moving faster in the eastward direction, but its downward velocity would not be altered. If the balloon were located 60 meters above the ground and was moving downward at 3 m/s, then it would take a time of 20 seconds to travel this vertical distance. d = v t So t = d / v = (60 m) / (3 m/s) = 20 seconds During the 20 seconds taken by the air balloon to fall the 60 meters to the ground, the wind would be carrying the balloon in the eastward direction. With a wind speed of 4 m/s, the distance traveled eastward in 20 seconds would be 80 meters. If the wind speed increased from the value of 4 m/s to a value of 6 m/s, then it would still take 20 seconds for the balloon to fall the 60 meters of downward distance. A motion in the downward direction is affected only by downward components of motion. An alteration in a horizontal component of motion (such as the eastward wind velocity) will have no affect upon vertical motion. Perpendicular components of motion are independent of each other. A variation of the eastward wind speed from a value of 4 m/s to a value of 6 m/s would only cause the balloon to be blown eastward a distance of 120 meters instead of the original 80 meters.

27. VECTOR RESOLUTION The process of determining the magnitude of a vector is called vector resolution. Two methods of vector resolution are: the parallelogram method the trigonometric method

28. PARALLELOGRAM METHOD A step-by-step procedure for using the parallelogram method of vector resolution is: Select a scale and accurately draw the vector to scale in the indicated direction. Sketch a parallelogram around the vector: beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a rectangle (a special case of a parallelogram).tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a rectangle (a special case of a parallelogram). Draw the components of the vector. The components are the sides of the parallelogram. The tail of the components start at the tail of the vector and stretches along the axes to the nearest corner of the parallelogram. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right). Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled F north. A rightward velocity component might be labeled v x ; etc. Measure the length of the sides of the parallelogram and use the scale to determine the magnitude of the components in real units. Label the magnitude on the diagram.use the scale to determine the magnitude of the components in real units. Label the magnitude on the diagram.

29. PARALLELOGRAM METHOD The following diagram illustrates the parallelogram method.

30. TRIGONOMETRIC METHOD Uses trig functions to determine the components of the vector. Used to determine the components of a single vector. Step by Step: Construct a rough sketch (no scale needed) of the vector in the indicated direction. Label its magnitude and the angle that it makes with the horizontal. Draw a rectangle about the vector such that the vector is the diagonal of the rectangle. Beginning at the tail of the vector, sketch vertical and horizontal lines. Then sketch horizontal and vertical lines at the head of the vector. The sketched lines will meet to form a rectangle.tail of the vector, sketch vertical and horizontal lines. Then sketch horizontal and vertical lines at the head of the vector. The sketched lines will meet to form a rectangle. Draw the components of the vector. The components are the sides of the rectangle. The tail of each component begins at the tail of the vector and stretches along the axes to the nearest corner of the rectangle. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right). Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled F north. A rightward force velocity component might be labeled v x ; etc. To determine the length of the side opposite the indicated angle, use the sine function. Substitute the magnitude of the vector for the length of the hypotenuse. Use some algebra to solve the equation for the length of the side opposite the indicated angle. Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

31. EXAMPLE OF TRIGONOMETRIC METHOD

32. COMPONENT METHOD OF VECTOR ADDITION Reminder you can add two vectors oriented at right angles to one another using the Pythagorean theorem. Example: Two displacement vectors with magnitude and direction of 11 km North and 11 km East can be added together to produce a resultant vector that is directed both North and East. The example below shows the two vectors added together head to tail. The resultant will be the hypotenuse of a right triangle. The sides of the triangle are both 11 km. Use the Pythagorean theorem to determine the hypotenuse side.

33. COMPONENT METHOD OF VECTOR ADDITION Using vector components, vector resolution, and the Pythagorean theorem to solve more complex vector addition problems. Adding three or more right angle vectors: Example 1: A student drives his car 6.0 km, North before making a right hand turn and driving 6.0 km to the East. Finally, the student makes a left hand turn and travels another 2.0 km to the north. What is the magnitude of the overall displacement of the student? Like any problem in physics, a successful solution begins with the development of a mental picture of the situation. The construction of a diagram like that below often proves useful in the visualization process.

34. COMPONENT METHOD OF VECTOR ADDITION When these three vectors are added together in head-to-tail fashion, the resultant is a vector that extends from the tail of the first vector (6.0 km, North, shown in red) to the arrowhead of the third vector (2.0 km, North, shown in green). The head-to-tail vector addition diagram is shown below.

35. COMPONENT METHOD OF VECTOR ADDITION As can be seen in the diagram, the resultant vector (drawn in black) is not the hypotenuse of any right triangle - at least not of any immediately obvious right triangle. But would it be possible to force this resultant vector to be the hypotenuse of a right triangle? The answer is Yes! To do so, the order in which the three vectors are added must be changed. The vectors above were drawn in the order in which they were driven. The student drove north, then east, and then north again. But if the three vectors are added in the order 6.0 km, N + 2.0 km, N + 6.0 km, E, then the diagram will look like this:

36. COMPONENT METHOD OF VECTOR ADDITION After rearranging the order in which the three vectors are added, the resultant vector is now the hypotenuse of a right triangle. The lengths of the perpendicular sides of the right triangle are 8.0 m, North (6.0 km + 2.0 km) and 6.0 km, East. The magnitude of the resultant vector (R) can be determined using the Pythagorean theorem. R 2 = (8.0 km) 2 + (6.0 km) 2 R 2 = 64.0 km 2 + 36.0 km 2 R 2 = 100.0 km 2 R = SQRT (100.0 km2) R = 10.0 km (SQRT indicates square root)

37. COMPONENT METHOD OF VECTOR ADDITION Example #2: Mac and Tosh are doing the Vector Walk Lab. Starting at the door of their physics classroom, they walk 2.0 meters, south. They make a right hand turn and walk 16.0 meters, west. They turn right again and walk 24.0 meters, north. They then turn left and walk 36.0 meters, west. What is the magnitude of their overall displacement? Diagram illustrates what is happening.

38. COMPONENT METHOD OF VECTOR ADDITION When these four vectors are added together in head-to-tail fashion, the resultant is a vector that extends from the tail of the first vector (2.0 m, South, shown in red) to the arrowhead of the fourth vector (36.0 m, West, shown in green). The head-to- tail vector addition diagram is shown below.

39. COMPONENT METHOD OF VECTOR ADDITION The resultant vector (drawn in black and labeled R) in the vector addition diagram above is not the hypotenuse of any immediately obvious right trangle. But by changing the order of addition of these four vectors, one can force this resultant vector to be the hypotenuse of a right triangle. For instance, by adding the vectors in the order of 2.0 m, S + 24.0 m, N + 16.0 m, W + 36.0 m. W, the resultant becomes the hypotenuse of a right triangle. This is shown in the vector addition diagram below.

40. COMPONENT METHOD OF VECTOR ADDITION With the vectors rearranged, the resultant is now the hypotenuse of a right triangle that has two perpendicular sides with lengths of 22.0 m, North and 52.0 m, West. The 22.0 m, North side is the result of 2.0 m, South and 24.0 m, North added together. The 52.0 m, West side is the result of 16.0 m, West and 36.0 m, West added together. The magnitude of the resultant vector (R) can be determined using the Pythagorean theorem. R 2 = (22.0 m) 2 + (52.0 m) 2 R 2 = 484.0 m 2 + 2704.0 m 2 R 2 = 3188.0 m 2 R = SQRT (3188.0 m2 2 ) R = 56.5 m (SQRT indicates square root)

41. SOH CAH TOA AND THE DIRECTION OF VECTORS To begin our discussion, let's return to Example 1 above where we made an effort to add three vectors: 6.0 km, N + 6.0 km, E + 2.0 km, N. In the solution, the order of addition of the three vectors was rearranged so that a right triangle was formed with the resultant being the hypotenuse of the triangle. The triangle is redrawn at the right. Observe that the angle in the lower left of the triangle has been labeled as theta (Θ). Theta (Θ) represents the angle that the vector makes with the north axis. Theta (Θ) can be calculated using one of the three trigonometric functions introduced earlier in this lesson - sine, cosine or tangent. The mnemonic SOH CAH TOA is a helpful way of remembering which function to use. In this problem, we wish to determine the angle measure of theta (Θ) and we know the length of the side opposite theta (Θ) - 6.0 km - and the length of the side adjacent the angle theta (Θ) - 8.0 km. The TOA of SOH CAH TOA indicates that the tangent of any angle is the ratio of the lengths of the side opposite to the side adjacent that angle. Thus, the tangent function will be used to calculate the angle measure of theta (Θ). The work is shown below.earlier in this lesson - sine, cosine or tangent. The mnemonic SOH CAH TOA is a helpful way of remembering which function to use. In this problem, we wish to determine the angle measure of theta (Θ) and we know the length of the side opposite theta (Θ) - 6.0 km - and the length of the side adjacent the angle theta (Θ) - 8.0 km. The TOA of SOH CAH TOA indicates that the tangent of any angle is the ratio of the lengths of the side opposite to the side adjacent that angle. Thus, the tangent function will be used to calculate the angle measure of theta (Θ). The work is shown below. Tangent(Θ) = Opposite/Adjacent Tangent(Θ) = 6.0/8.0 Tangent(Θ) = 0.75 Θ = tan -1 (0.75) Θ = 36.869 …° Θ =37°

42. ADDITION OF NON PERPENDICULAR VECTORS Use concept of a vector component and the process of vector resolution. A vector component describes the effect of a vector in a given direction. Any angled vector has two components; one is directed horizontally and the other is directed vertically. For instance, a northwest vector has a northward component and a westward component. Together, the effect these two components are equal to the overall effect of the angled vector.

43. ADDITION OF NON PERPENDICULAR VECTORS Example – A plane flies northwest from Chicago towards the Canadian border. The northwest displacement vector of the plane has two components ( A northward component and a westward component). When added together, these two components are equal to the overall northwest displacement.

44. ADDITION OF NON PERPENDICULAR VECTORS The northwest vector has north and west components that are represented as A x and A y. It can be said that A = A x + A y So whenever we think of a northwest vector, we can think instead of two vectors - a north and a west vector. The two components A x + A y can be substituted in for the single vector A in the problem. Now suppose that your task involves adding two non-perpendicular vectors together. We will call the vectors A and B. Vector A is a nasty angled vector that is neither horizontal nor vertical. And vector B is a nice, polite vector directed horizontally. The situation is shown below.

45. ADDITION OF NON PERPENDICULAR VECTORS Of course nasty vector A has two components - A x and A y. These two components together are equal to vector A. That is, A = A x + A y. And since this is true, it makes since to say that A + B = A x + A y + B.

46. ADDITION OF NON PERPENDICULAR VECTORS And so the problem of A + B has been transformed into a problem in which all vectors are at right angles to each other. Nasty has been replaced by nice and that should make any physics student happy. With all vectors being at right angles to one another, their addition leads to a resultant that is at the hypotenuse of a right triangle. The Pythagorean theorem can then be used to determine the magnitude of the resultant.

47. BIBLIOGRAPHY The Physics Classroom. (2013). “Vectors – Fundamentals and operations”. Retrieved from http://www.physicsclassroom.com/Class/vectors/u3l1a.cfmhttp://www.physicsclassroom.com/Class/vectors/u3l1a.cfm All images are from The Physics Classroom website.