Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position From the Toolbox (what you need from previous classes): Cartesian Coordinates: Coordinates of points in general, P = (x, y) or P = (x, y, z); cooridinates (x, f (x)) of points on the graph of a function. Trigonometry: Polar coordinates (x, y) = (r cos θ, r sin θ). Vectors: Sketching a vector using its component form. Vector equations of lines. A vector function is a function whose input is a scalar (called the parameter) and whose output is a vector. An important application of vector functions is describing motion. The output vector indicates the position of an object at time t. In this worksheet, you will: Match curves to vector functions that parameterize them. Use vector functions to find locations on a curve based on parameter values, and to find parameter values corresponding to particular locations. Analyze motion (position, relative speed, direction of travel, etc) based on vector functions. Use a CAS or graphing app to graph parametric curves. For this worksheet, you may find the following helpful: A graphing app for vector functions. Some suggestions can be found on the course website: http://math.boisestate.edu/ultman275/links/ Notes: Common Parameterizations for Some Important Curves http://math.boisestate.edu/ shariultman/275/paramcurves.pdf (You may use these notes during exams.)
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position 1 Model 1: LINES AND POSITION FUNCTIONS Diagram 1A r 1 (t) = 3t 1, 3t + 4 r 1 (0) = 3(0) 1, 3(0) + 4 = 1, 4 r 1 (1) = 3(1) 1, 3(1) + 4 = 2, 1 Diagram 1B r 2 (t) = 2t + 1, 2t + 2 r 2 (0) = 2(0) + 1, 2(0) + 2 = 1, 2 r 2 (1) = 2(1) + 1, 2(1) + 2 =, Diagram 1C r 3 (t) = 5t + 5, 5t 2 r 3 (0) = 5( ) + 5, 5( ) 2 = 5, 2 r 3 (1) =, Critical Thinking Questions In this section, you will compare different vector functions parameterizing the same line, and see how they can be used to describe the motion of three different objects.
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position 2 The three vector equations in Model 1 describe the position at time t (given in seconds) for three different objects traveling along the same line. (Q1) The position of Object 1 at t seconds is given by the vector function r 1 (t). (a) On Diagram 1: Sketch the vector r 1 (1) that gives the position of Object 1 at time t = 1 seconds. (b) What are the locations (as points P 0 and P 1 ) of Object 1 at t = 0 seconds and t = 1 second? P 0 = P 1 = (Q2) The position of Object 2 at t seconds is given by the vector function r 2 (t). (a) On Diagram 2: Find the components of the vector r 2 (1) that gives the position of Object 2 at time t = 1 seconds, and sketch this vector. (b) What are the locations (as points P 0 and P 1 ) of Object 2 at t = 0 seconds and t = 1 second? P 0 = P 1 = (Q3) The position of Object 3 at t seconds is given by the vector function r 3 (t). (a) On Diagram 3, find the components of the vectors r 3 (0) and r 3 (1) that give the positions of Object 3 at times t = 0 and t = 1 seconds, and sketch the vector r 3 (1). (b) What are the locations (as points P 0 and P 1 ) of Object 3 at t = 0 seconds and t = 1 second? P 0 = P 1 = (Q4) Which (if any) of the objects are traveling along the line from the bottom right to the top left? Object 1 Object 2 Object 3 (Q5) Which object crosses the x-axis in the shortest amount of time after t = 0 seconds, and how much time elapses for this object? (Q6) If t = 0 marks the present time, which (if any) objects have crossed the x-axis in the past, and at what time(s)? (Q7) Which object is traveling the fastest (covering the most distance per second) between t = 0 and t = 1 seconds? Object 1 Object 2 Object 3
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position 3 ( Q8) Find a parameterization (vector function) that describes the position of an object traveling along the line in Model 1, crossing the y-axis at exactly t = 0 seconds, and crossing the x-axis at exactly t = 1 second. ( Q9) Find a parameterization (vector function) that describes the position of an object moving in 3-space along a line parallel to the z-axis, meeting the following two conditions: it passes through the xy-plane at the point P 0 = (2, 3, 0) at exactly t = 0 seconds, and it passes through the point P 1 = (2, 3, 5) at exactly t = 1 seconds. Model 2: Different Ways of Parameterizing a Parabola Diagram 2A Diagram 2B Diagram 2C r 1 (t) = t, t 2, < t < r 2 (t) = t, t 2, 0 t < r 3 (t) = t, t 2, 1 t 1 r 4 (t) = t, t, 0 t < Critical Thinking Questions In this section, you will compare different ways of parameterizing the parabola y = x 2.
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position 4 (Q10) Each of the vector functions in Diagram 2 parameterize either the full parabola y = x 2, or a section of the parabola, as pictured in Diagrams 2A, 2B, and 2C. Which of the vector functions parameterize the full parabola, pictured in Diagram 2A? r 1 (t) r 2 (t) r 3 (t) r 4 (t) (Q11) Which of the vector functions parameterize the right half of the parabola, pictured in Diagram 2B? r 1 (t) r 2 (t) r 3 (t) r 4 (t) (Q12) Which of the vector functions parameterize the bottom section of the parabola, pictured in Diagram 2C? r 1 (t) r 2 (t) r 3 (t) r 4 (t) (Q13) The vector functions r 1 (t), r 2 (t), and r 3 (t) in Model 2 look very similar. Explain why, despite their similarity, they parameterize different parts of the parabola. (Q14) Suppose the vector functions in Model 2 describe the position of objects as they traveled along parabolic paths. These objects would share a common direction of travel as time t increases. Describe this direction of travel. ( Q15) In Model 2, you can tell that all four vector functions r i (t) = x(t), y(t) parameterize all or part of the parabola y = x 2 by looking at the relationship between the x- and y-components. For each of the four vector functions r i (t) = x(t), y(t), you see that: the curve is the parabola: y = x 2 the components of r i (t) = x(t), y(t) are related by: y(t) = [ x(t) ] 2. For example, r 1 (t) = t, t 2, so x(t) =, y(t) = [ x(t) ] 2 =. For r 4 (t) = t, t, x(t) =, y(t) = [ x(t) ] 2 =. Now, find vector functions that parameterize the graphs of the functions y = x 3, y = e x, and x = cos y.
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position 5 ( Q16) The section of parabola pictured in Diagram 2B is parameterized by r 2 (t). It can also be parameterized by the vector function r(t) = t 2, t 4, t. This new parameterization behaves differently than r 1 (t). Describe this difference by looking at the directions of travel for < t 0 and 0 t <. What happens when t = 0? ( Q17) Consider the vector function r(t) = sin t, sin 2 t, t. Identify which of the three curves (Diagram 2A, 2B, or 2C) this vector function parameterizes, and describe how an object with this position function would move along the curve. Model 3: Circles, Ellipses, and Helices Diagram 3 Positions of five different objects at time t: r 1 (t) = 2 cos t î + 2 sin t ĵ r 2 (t) = 2 cos(3t) î + 2 sin(3t) ĵ r 3 (t) = 2 sin t î + 2 cos t ĵ r 4 (t) = 3 cos t î + 3 sin t ĵ r 5 (t) = 3 cos( t) î + 3 sin( t) ĵ Critical Thinking Questions In this section, you will compare different ways of parameterizing a circle. (Q18) In Diagram 3: What is the radius of the smaller circle? What is the radius of the larger circle?
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position 6 (Q19) Compute the magnitudes for each of the five vector functions in Diagram 3: Vector Function Magnitude r 1 (t) = 2 cos t î + 2 sin t ĵ r 1 (t) = 4 cos 2 t + 4 sin 2 t = 2 r 2 (t) = 2 cos(3t) î + 2 sin(3t) ĵ r 2 (t) = cos 2 (3t) + sin 2 (3t) = r 3 (t) = 2 sin t î + 2 cos t ĵ r 3 (t) = sin 2 t + cos 2 t = r 4 (t) = 3 cos t î + 3 sin t ĵ r 4 (t) = cos 2 t + sin 2 t = r 5 (t) = 3 cos( t) î + 3 sin( t) ĵ r 5 (t) = cos 2 ( t) + sin 2 ( t) = A circle x 2 + y 2 = R 2 (sets of points at a constant distance R from the origin) is parameterized by a vector function with constant magnitude, r(t) = R. (Q20) Which of the vector functions listed in Diagram 3 parameterize the circle of radius r = 2? r 1 (t) r 2 (t) r 3 (t) r 4 (t) r 5 (t) (Q21) Which of the vector functions listed in Diagram 3 parameterize the circle of radius r = 3? r 1 (t) r 2 (t) r 3 (t) r 4 (t) r 5 (t) (Q22) Which one of the vector functions listed in Diagram 3 describe the motion of the object traveling the fastest around the circle of radius r = 2? r 1 (t) r 2 (t) r 3 (t) r 4 (t) r 5 (t) (Q23) What is the location of the object whose position is given by r 1 (t) at the two times t = 0 and t = π/2? Recall: cos 0 = 1, sin 0 = 0, cos π/2 = 0, sin π/2 = 1. (Q24) What is the location of the object whose position is given by r 3 (t) at the two times t = 0 and t = π/2? ( Q25) What is the direction of travel of each of the objects whose positions are given by the vector functions in Diagram 3? r 1 (t) r 2 (t) r 3 (t) r 4 (t) r 5 (t) cws / ccws cws / ccws cws / ccws cws / ccws cws / ccws cws = clockwise, ccws = counterclockwise
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position 7 ( Q26) Construct a vector function that describes the motion of an object traveling counterclockwise around the circle x 2 + y 2 = 25, located at the point P = ( 5, 0) when t = 0. You will need a graphing app for curves in the plane (R 2 ) and 3-space (R 3 ) to explore the behavior of vector functions parameterizing the following curves: Circles centered at the origin: r(t) = R cos t î + R sin t ĵ Ellipses in standard position: r(t) = a cos t î + b sin t ĵ Helices: r(t) = R cos t î + R sin t ĵ + bt ˆk ( Q27) Identify the common parts of the three vector equations given in Model 4 describing circles, ellipses, and helices. ( Q28) Identify the difference between the vector equations given in Model 4 describing circles and ellipses. ( Q29) Identify the difference between the vector equations given in Model 4 describing circles and helices. ( Q30) Using your app, graph the circle r(t) = 3 cos t î + 3 sin t ĵ. Now, leaving the 3 cos t î as it is, vary the coefficient for the sine function (for exmple, graph r(t) = 3 cos t î + 2 sin t ĵ, r(t) = 3 cos t î + 4 sin t ĵ, etc.), and observe how this changes the curve. ( Q31) Using your app, graph the helix r(t) = 3 cos t î + 3 sin t ĵ + t ˆk. Describe the shape of the resulting curve. Rotate your graph so you are looking down along the z-axis. What curve results from projecting the helix into the xy-plane? ( Q32) Graph helices with different constant coefficients in the ˆk -component (for example, r(t) = 3 cos t î + 3 sin t ĵ + 2t ˆk, r(t) = 3 cos t î + 3 sin t ĵ + 1 2 t ˆk ). Observe how changing the coefficient in the ˆk -component changes the shape of the helix. What happens if you graph r(t) = 3 cos t î + 3 sin t ĵ + t 2 ˆk? ( Q33) Graph the circle r(t) = cos t, sin t, then graph the circle with the constant vector C = 1, 2 added: r(t) = cos t, sin t + 1, 2 = cos t + 1, sin t + 2. Do the same for the parabola r(t) = t, t 2 and the parabola with a constant vector C = 1, 2 added: r(t) = t, t 2 + 1, 2 = t +1, t 2 +2. In general, what happens to the graph of a curve if you add a constant vector?