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Vector Presentation

Transcript: Vector mapping project Intro We used the Belknap academic building and ekstrom library as our two locations. we measured the vectors using skills learned from this class . Path 1 Path 1 The first path we established, labeled 1 with each vector associated by alphabetical character. Leaving the Academic building from the North-Eastern door and walking a straight path for 613.48 ft using Google Maps set to view a satellite image set to north being the y-axis. We took a screen shot on the path and posted it to our One Note file to use its compass to measure the angle of the first vector in Path 1, we labeled this vector 1a. At the end of vector 1a we measured the distance for 1b. At the tip of vector 1a begins the origin of 1b. Path 1b is a vector with magnitude of 906.38 ft at an angle of 172 degrees measured using North as a y-axis on Google Maps. Finally, Vector 1c of Path 1 was oriented at the end of vector 1b. Using North on google maps as a reference of the y-axis we measured 1c to have a magnitude of 328.58 ft at an angle of 270 degrees. Path 2 path 2 Our second path, labeled Path 2 took what we considered to be the more direct route. Both paths 1 and 2 started at what we considered the same origin. We felt that using different doors was acceptable considering that the whole building was what we were call this origin. Starting Path 2 began at the western entrance to the building. Using the method of measuring distance and angles as described previously, vector 2a had a magnitude of 413.27 ft and measuring angle of 172 degrees referencing north on google maps as our y-axis. Vector 2b begins where vector 2a ended, with a magnitude of 306.50 ft and an angle of 83 degrees. Vector 2c the final vector of Path 2 measured at an angle of 154 degrees and with magnitude of 214.84 ft. programming programming Our team developed a program using the Python Idle 3.9.1 to compute the data that we recorded. We first needed to let the program know that we were going to be asking it to do some calculations. So, we used the “import math” to allow it to understand and help us with calculations. Now that the program understand the math functions we'll be using, we added input for the user to identify the type of calculation they would like to perform. Now we needed to let our program know how to keep a memory of our range of possible inputs. Using the “tmpList” function gave our program a memory. This allowed the user to input as many vectors as they wished for the program to convert. Our “If-else if” statement allowed the user to input and assign variables to what we were later going to ask the program to compute. By assigning inputs with a float attached gives the user the ability to use decimals when defining variables. Finally, in both the “if” and the “elif” statements end using a “tmpList.append (xxxx)” where the xxxx is what we asked the program to remember and how to format it. Programming (cont.) programming (cont.) Now that we have asked the program to do a task, we need to teach it what we mean when asking to perform said task. With out being specific in each of these functions we could confuse the program and have errors. When writing a definition of a function we needed to give the function a through function that lets the program know what inputs to use in the function that we defined. “displace = math.sqrt((tmpX ** 2) + (tmpY ** 2))” this is simply the Pythagorean Theorem, a great example of the difference between how we usually use math and how our program sees math. programming (cont.) Programming (cont.) To review, we have shown the program how to ask the question that we are teaching it and we have defined all functions and variable its needs to solve the questions. This is our “main” function. Our main function is the work horse of the program. In the picture, the main function is shown as a visual aid of all the functions we defined, how we told the program to compute each function, and how to associate the functions. Also shown in the figure is how we added in text to explain to the user what exactly was computed. This helps clarify the results. Programming (cont.) programming (cont.) Once we finished our coding, we tested it to show that it was working properly. Using our understanding of Polar and Cartesian coordinates we asked the program to perform a up 1 unit and left 1 unit, using vector addition and the definition Pythagorean Theorem. To prove this is the accurate calculation we checked that the program answered with the decimal value of the equation we used that’s equal to 1.414. We asked the program to round the second decimal place using the round function (xxxx, 2); where xxxx is the variable we asked it to round for us. The program was behaving correctly, we explained to it we were going to ask it a range of questions, and taught it how to handle all possible inputs. Now to show our program performing the exact calculations. first successful test. exact calculations Excel excel After we mapped

Vector Presentation

Transcript: Named after it's discoverer, Pythagoras, it is one of the earliest theorems known to civilization. It is a part of Euclidean geometry and deals with only right triangles. The definition of the Pythagorean theorem is "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." Basically, a^2 +b^2 = c^2 Cosine The cosine function is used to find the x component of the vetor. Cos = adj/hyp adj is the adjacent side of the angle . hyp is the hypotenuse. As with the sine function we will have to rearrange the equation to find the adjacent side (x componet). The equation will look like: (hyp)Cos = adj Sine The sine function is used to find the y component of the vector. Sin = opp/hyp opp is the opposite side of the angle . hyp is the hypotenuse side of the triangle. When you are given a vector you are given the angle and the hypotenuse. You need to find the opposite side (y component). In order to do this we will have to rearrange the equation like so: (hyp)Sin = opp 35º Adding Non-Perpendicular Vectors One of the easiest and quickest ways to define a vector is to use the Pythagorean theorem and the tangent function. After you use the tail to head method to graph your vectors you connect the tail of the first to the head of the last. Now, using the Pythagorean theorem, you square your adjacent and opposite sides, add them together, find the square root of the sum, and you have the magnitude of the resultant. Next, you use the tangent function and you have the heading of the resultant. Remember that tan=opp/adj. Unfortunately none of this will work if the vectors do not form a right triangle. Vector Operations Example Given: v=95 km/h =35º Unknown: vx=? vy=? Rearranged equations: vy = vsin vx = vcos vy = (95 km/h)(sin35º) vy = 54km/h vx = (95km/h)(cos35º) vx = 78km/h What happens when you are given a resultant vector and you need to fingure out its components? In order to find the components you have to resolve the vector. Resolving the vector breaks the vector down into its x and y components. But how do you do this? Vectors When adding non-perpendicular vectors you must break each vector into its components. What this means is that you must break each vector into its x and y components. After you have broken each vector into its component you can add their x and y values together. Plot the resulting x and y coordinates and now you can use the Pythagorean theorem and tangent function to find the resultant. vy Tangent is defined as opposite/adjacent. It is very useful because you can enter the side lengths of two sides and find the degree of any angle, depending on which two side lengths you enter into the equation. When making a coordinate system you plot two different values along the x and y axis respectivly. A common coordinate system is a compass. It uses the y axis as north and south and the x axis as west to east. v = 95km/h vx Resolving Vectors Tangent Function There are two equations used to find the components. Sin = opp/hyp Cos = adj/hyp These are the trig funtctions sine and cosine. Using the theorem and tangent The Pythagorean Theorem Coordinate systems In order to solve a problem involving coordinate systems you have to use one that can actually solve the problem. For example if you were trying to measure the velocity of a car you couldn't use a coordinate system that measured altitude. Using sine and cosine To find the x component of a vector you need to use cos. You multiply the magnitude of the vector by the cos of the degree of the vector. Finding the y component is just as easy, you just swap sin in for cos. Abbreviated sin and cos

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