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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

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.6: Parametric Equations
Problem 97E
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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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