Motion in Space: Compute tangent vectors to parametric curves; solve problems involving velocity, speed, and acceleration. Consider the standard helix parameterized by r(t) =< cos(t),sin(t),t>. a. Find the unit tangent vector >T(t) to r(t). b. Explain why the fact that T(t) has length 1 implies that T and T 'are orthogonal vectors for every value of t. (Hint: note that 'T ·'T = |||T||2, and compute d/dt(>T.>T>) c. For the given function r) with unit tangent vector &T you found in (a), determine N(t) = 1 ||'T '(t)|| \T '(t). d. What geometric properties does 'N(t) have? That is, what is its length and what is the angel between T and >N?
Motion in Space: Compute tangent vectors to parametric curves; solve problems involving velocity, speed, and acceleration. Consider the standard helix parameterized by r(t) =< cos(t),sin(t),t>. a. Find the unit tangent vector >T(t) to r(t). b. Explain why the fact that T(t) has length 1 implies that T and T 'are orthogonal vectors for every value of t. (Hint: note that 'T ·'T = |||T||2, and compute d/dt(>T.>T>) c. For the given function r) with unit tangent vector &T you found in (a), determine N(t) = 1 ||'T '(t)|| \T '(t). d. What geometric properties does 'N(t) have? That is, what is its length and what is the angel between T and >N?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Motion in Space: Compute tangent
vectors to parametric curves; solve
problems involving velocity, speed, and
acceleration. Consider the standard helix
parameterized by r(t) =< cos(t),sin(t),t>.
a. Find the unit tangent vector >T(t) to
r(t).
b. Explain why the fact that T(t) has
length 1 implies that T and 'T 'are
orthogonal vectors for every value of t.
(Hint: note that 'T ·'T = |||T||2, and
compute d/dt( >T.>T>)
c. For the given function r) with unit
tangent vector T you found in (a),
determine N(t) = 1 ||'T '(t)|| \T '(t).
d. What geometric properties does >N(t)
have? That is, what is its length and what
is the angel between T and 'N?
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