Math 1172 Project #4 Applications of Vectors, Coordinate Transformations, and Parameterization by Arclength Due: Thursday, April 13 Name(s): - - - - - - - - - - - - - - - - - - - - Description - - - - - - - - - - - - - - - - - - - Vectors are one of the most important tools in multivariable calculus and in the physical sciences. Many problems in coordinate geometry can be solved nicely by the introducing vectors and utilizing the geometric properties of dot and cross products. Many problems in the physical sciences have a natural implied symmetry. Gravitational force between two objects, for instance, depends only on the distance between them; thus, any point that is a fixed distance r from an object will experience the same gravitational force! In these instances, it is advisable to work in polar coordinates rather than Cartesian coordinates. Of course, there are other types of symmetry that suggest certain coordinates be used. The coordinates discussed in this project introduce the hyperbolic polar coordinates that arise frequently in special relativity. Finally, arclength is a necessary theoretical parameter. Often, we want to discuss how quantities on a parametrically defined curve change on the infinitesimal level, and the only way to ensure that small changes in the parameter do not lead to large distances travelled along the curve is to parameterize by arclength. You will explore many of these scenarios in your next course in multivariable calculus, but we preview one important application here. - - - - - - - - - - - - - - Purpose of the Assignment - - - - - - - - - - - - - - To show how to solve a plane geometry problem by introducing vectors and utilizing projections. To explore calculus for a curvilinear coordinate system other than polar coordinates. To show the importance of arclength as a theoretical parameter. - - - - - - - - - - - - - - - - - - - - - Directions - - - - - - - - - - - - - - - - - - - - This assignment is worth 15 pts. You are STRONGLY encouraged to work in groups of up to 3 students with the following restraints: - The students in your group must have the same recitation instructor. - Each group will submit one copy of this assignment; group members should NOT submit individual assignments! - Each group member's name should appear on the top of this page. - Each member of the group will receive the same grade. You may of course work with students outside of your recitation instructor, but they must hand in their own version of the assignment to their instructor. If you need more space than what is provided, feel free to use extra sheets pf paper. For work on extra sheets to be considered, you must staple them to your assignment, clearly indicate to which problem any work belongs, and restaple the assignment! Problem 1 : [5 pts] (Distance Between a Point and A Line) The distance, d, between a point P = (x0 , y0 ) and a line l is given by: np o 2 2 d := min (x x0 ) + (y y0 ) (x,y)l where (x, y) is a point on the line. In other words, the distance between a point and a line is defined to be the smallest of all of the distances between the given point and a point on the line. The following exercise will guide you to figure out how to find this distance for a particular example. I. Suppose that we wish to find the distance between the point (4, 1) and y = 2x. A. Show that the given point is not on the line y = 2x. B. Do the following on the axes provided. Plot the point (4, 1) on the axes. Draw a line segment perpendicular to the line y = 2x that extends to the point (4, 1). Label it d. The length of this line segment is the distance between (4, 1) and a certain point on the line. argue why this must be the shortest distance between (4, 1) and any point on the line as follows: Pick any other point on the line and label it in your picture. Draw a line segment connecting this point to (4, 1). C. Using your picture, explain clearly why d must be less than the length of the segment you drew! D. Our goal is to use the techniques we have involving dot products in order to find d. In order to formulate this problem in a way that allows us to do this, do the following to the axes provided below. Plot the point (4, 1) on the axes as a vector. Label it ~u. Draw a vector perpendicular to the line y = 2x that extends to the point (4, 1). ~ . It should be clear that the magnitude of N ~ is d! Label it N Draw a vector along the line y = 2x that extends from the origin to the point on ~ begins. Label it P~ . the line where N E. Explain how P~ relates to ~u. Then, calculate P~ . ~ ? Use this to find N ~. F. How does ~u relate to P~ + N ~ |, calculate d. G. Since d = |N Congratulations! You just found the distance between the line y = 2x and the point (4, 1)! Note that if the line originally does not pass through the origin, this technique does not immediately apply. If you want to find the distance between the point (x0 , y0 ) and the line y = mx + b, you can simply shift the picture to one of the type treated here by finding the distance between the point (x0 , y0 b) and y = mx. Note this does NOT change the actual distance; it simply shifts the point and line to a coordinate system where the line passes through the origin and the point is translated appropriately! Problem 2 : [5 pts] (Hyperbolic Coordinates) The polar coordinates (r, ) provide a different way to describe a point P in the xy-plane from its usual Cartesian description (x, y). Given a description (r, ) for P, one can calculate the Cartesian description for P using the relationships: ( x = r cos y = r sin There are of course other ways in which the point P can be described. One such way is to introduce the hyperbolic coordinate system. First, we define the hyperbolic trigonometric functions 1 cosh x and sinh x by: ez + ez cosh z = 2 z z sinh z = e e 2 Now, we may define the hyperbolic coordinates (R, ) of a given point P in the xy-plane by requiring that the Cartesian description (x, y) of P be given by the relationships: ( x = R cosh y = R sinh Hyperbolic coordinates are fundamentally important in special relativity and hyperbolic geometries are of interest in higher mathematics. 1 There is a reason that these are called \"trigonometric functions\