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55. Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y² and z = x² at the point (1, 1, 1).

55. Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y² and z = x² at the point (1, 1, 1).

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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lating circles and the parabola on the same screen. 53. At what point on the curve x = t³, y = 3t, z = t¹ is the normal plane parallel to the plane 6x + 6y - 8z = 1? CAS 54. Is there a point on the curve in Exercise 53 where the osculating plane is parallel to the plane x + y + z = 1? [Note: You will need a CAS for differentiating, for simplify- ing, and for computing a cross product.] 55. Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y² and Z = = x² at the point (1, 1, 1). slabog 9255 56. Show that the osculating plane at every point on the curve r(t) = (t + 2, 1 t, 1t²) is the same plane. What can you conclude about the curve? - 57. Show that at every point on the curve r(t) = (e' cos t, e' sin t, e¹) the angle between the unit tangent vector and the z-axis is the same. Then show that the same result holds true for the unit normal and binormal vectors. 58. The rectifying plane of a curve at a point is the plane that contains the vectors T and B at that point. Find the recti- fying plane of the curve r(t) = sin ti + cos tj + tan t k at the point (√√2/2, √2/2, 1). 59. Show that the curvature K is related to the tangent and normal vectors by the equation dT ds = KN BAUDIA
Transcribed Image Text:lating circles and the parabola on the same screen. 53. At what point on the curve x = t³, y = 3t, z = t¹ is the normal plane parallel to the plane 6x + 6y - 8z = 1? CAS 54. Is there a point on the curve in Exercise 53 where the osculating plane is parallel to the plane x + y + z = 1? [Note: You will need a CAS for differentiating, for simplify- ing, and for computing a cross product.] 55. Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y² and Z = = x² at the point (1, 1, 1). slabog 9255 56. Show that the osculating plane at every point on the curve r(t) = (t + 2, 1 t, 1t²) is the same plane. What can you conclude about the curve? - 57. Show that at every point on the curve r(t) = (e' cos t, e' sin t, e¹) the angle between the unit tangent vector and the z-axis is the same. Then show that the same result holds true for the unit normal and binormal vectors. 58. The rectifying plane of a curve at a point is the plane that contains the vectors T and B at that point. Find the recti- fying plane of the curve r(t) = sin ti + cos tj + tan t k at the point (√√2/2, √2/2, 1). 59. Show that the curvature K is related to the tangent and normal vectors by the equation dT ds = KN BAUDIA
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