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DISCUSS: Same Line: Different Parametric Equations Every line can be described by infinitely many different sets of parametric equations, since any point on the line and any vector parallel to the line can be used to construct the equa- tions. But how can we tell whether two sets of parametric equations represent the same line? Consider the following two sets of parametric equations: Line 1: x = – t, y= 31, z = -6 + 5t Line 2: x = -1 + 21, y = 6 – 6t, z = 4 – 10t (a) Find two points that lie on Line 1 by setting t = 0 and 1 = 1 in its parametric equations. Then show that these points also lie on Line 2 by finding two values of the parameter that give these points when substituted into the parametric equations for Line 2. (b) Show that the following two lines are not the same by finding a point on Line 3 and then showing that it does not lie on Line 4. Line 3: x= 41, y= 3 – 61, z = -5 + 21 Line 4: x = 8 – 21, y = -9 + 3t, z = 6 - t

DISCUSS: Same Line: Different Parametric Equations Every line can be described by infinitely many different sets of parametric equations, since any point on the line and any vector parallel to the line can be used to construct the equa- tions. But how can we tell whether two sets of parametric equations represent the same line? Consider the following two sets of parametric equations: Line 1: x = – t, y= 31, z = -6 + 5t Line 2: x = -1 + 21, y = 6 – 6t, z = 4 – 10t (a) Find two points that lie on Line 1 by setting t = 0 and 1 = 1 in its parametric equations. Then show that these points also lie on Line 2 by finding two values of the parameter that give these points when substituted into the parametric equations for Line 2. (b) Show that the following two lines are not the same by finding a point on Line 3 and then showing that it does not lie on Line 4. Line 3: x= 41, y= 3 – 61, z = -5 + 21 Line 4: x = 8 – 21, y = -9 + 3t, z = 6 - t

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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DISCUSS: Same Line: Different Parametric Equations Every line can be described by infinitely many different sets of parametric equations, since any point on the line and any vector parallel to the line can be used to construct the equa- tions. But how can we tell whether two sets of parametric equations represent the same line? Consider the following two sets of parametric equations: Line 1: x = – t, y= 31, z = -6 + 5t Line 2: x = -1 + 21, y = 6 – 6t, z = 4 – 10t (a) Find two points that lie on Line 1 by setting t = 0 and 1 = 1 in its parametric equations. Then show that these points also lie on Line 2 by finding two values of the parameter that give these points when substituted into the parametric equations for Line 2. (b) Show that the following two lines are not the same by finding a point on Line 3 and then showing that it does not lie on Line 4. Line 3: x= 41, y= 3 – 61, z = -5 + 21 Line 4: x = 8 – 21, y = -9 + 3t, z = 6 - t
Transcribed Image Text:DISCUSS: Same Line: Different Parametric Equations Every line can be described by infinitely many different sets of parametric equations, since any point on the line and any vector parallel to the line can be used to construct the equa- tions. But how can we tell whether two sets of parametric equations represent the same line? Consider the following two sets of parametric equations: Line 1: x = – t, y= 31, z = -6 + 5t Line 2: x = -1 + 21, y = 6 – 6t, z = 4 – 10t (a) Find two points that lie on Line 1 by setting t = 0 and 1 = 1 in its parametric equations. Then show that these points also lie on Line 2 by finding two values of the parameter that give these points when substituted into the parametric equations for Line 2. (b) Show that the following two lines are not the same by finding a point on Line 3 and then showing that it does not lie on Line 4. Line 3: x= 41, y= 3 – 61, z = -5 + 21 Line 4: x = 8 – 21, y = -9 + 3t, z = 6 - t
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