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a. Show that n(t) = - gʻ(t)i + f'(t)j and - n(t) = g'(t)i – f'(t)j are both normal to the curve r(t) = f(t)i + g(t)j at the point (f(t),g(t)). To obtain N for a particular plane curve, choose the one of n or -n that points toward the concave side of the curve, and make it into a unit vector. (See the figure to the right.) Apply this method to find N for the following curves. P. b. r(t) = ti + 3 3 e'j 3 3 Pot K ds c. r(t) = /9 – 16? i + 4tj, -sts- The vector dT/ds, normal to the curve, always points in the direction in which T is turning. The unit normal vector N is the direction of dT/ds.

a. Show that n(t) = - gʻ(t)i + f'(t)j and - n(t) = g'(t)i – f'(t)j are both normal to the curve r(t) = f(t)i + g(t)j at the point (f(t),g(t)). To obtain N for a particular plane curve, choose the one of n or -n that points toward the concave side of the curve, and make it into a unit vector. (See the figure to the right.) Apply this method to find N for the following curves. P. b. r(t) = ti + 3 3 e'j 3 3 Pot K ds c. r(t) = /9 – 16? i + 4tj, -sts- The vector dT/ds, normal to the curve, always points in the direction in which T is turning. The unit normal vector N is the direction of dT/ds.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 18T
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Find B and C 

 

c. N= Di+C
Transcribed Image Text:c. N= Di+C
a. Show that n(t) = - g'(t)i + f' (t)j and - n(t) = g'(t)i -f (t)j are both normal to the curve r(t) = f(t)i + g(t)j at the point (f(t),g(t)). To obtain N for a particular plane curve, choose the one of n or -n that points toward the concave side of the curve, and make it into a unit vector. (See the figure to the right.) Apply this method to find N for the following curves. K ds T. b. r(t) = ti +3 e'i 3 Pot N -. .2 c. r(t) = /9 – 16ti+ 4tj, sts %3D 4 The vector dT/ds, normal to the curve, always points in the direction in which T is turning. The unit normal vector N is the direction of dT/ds. a. To show that n(t) and - n(t) are both normal to r(t), first find v(t). v(t) = f'(t) i+ g'(t) j To show that n is normal to r, show that n•T=0. If n is normal to r, then so is - n, because the components of -n are the same as those of n with sign changed. How are v and T related? The vector v is parallel to T. How can v(t) be used to show that vector n is normal to the curve r(t)? A. Show that v•T= - 1. B. Show that v•T= 0. C. Show that n•v = - 1. D. Show that n•v = 0. Why is the equation from the previous step satisfied? A. The components of n(t) are negative reciprocals of the components of T, so the dot product is - 1. B. The sum of the components of v(t) is the negative of T, so the dot product is 0. C. The components of n(t) are negative reciprocals of the components of v(t), so the dot product is -1. 'D. The components of n(t) are the components of v(t) with the order swapped and the sign of one changed, so the dot product is 0. b. N = i+
Transcribed Image Text:a. Show that n(t) = - g'(t)i + f' (t)j and - n(t) = g'(t)i -f (t)j are both normal to the curve r(t) = f(t)i + g(t)j at the point (f(t),g(t)). To obtain N for a particular plane curve, choose the one of n or -n that points toward the concave side of the curve, and make it into a unit vector. (See the figure to the right.) Apply this method to find N for the following curves. K ds T. b. r(t) = ti +3 e'i 3 Pot N -. .2 c. r(t) = /9 – 16ti+ 4tj, sts %3D 4 The vector dT/ds, normal to the curve, always points in the direction in which T is turning. The unit normal vector N is the direction of dT/ds. a. To show that n(t) and - n(t) are both normal to r(t), first find v(t). v(t) = f'(t) i+ g'(t) j To show that n is normal to r, show that n•T=0. If n is normal to r, then so is - n, because the components of -n are the same as those of n with sign changed. How are v and T related? The vector v is parallel to T. How can v(t) be used to show that vector n is normal to the curve r(t)? A. Show that v•T= - 1. B. Show that v•T= 0. C. Show that n•v = - 1. D. Show that n•v = 0. Why is the equation from the previous step satisfied? A. The components of n(t) are negative reciprocals of the components of T, so the dot product is - 1. B. The sum of the components of v(t) is the negative of T, so the dot product is 0. C. The components of n(t) are negative reciprocals of the components of v(t), so the dot product is -1. 'D. The components of n(t) are the components of v(t) with the order swapped and the sign of one changed, so the dot product is 0. b. N = i+
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