Find the tangent vector Tbar(t) and a set of parametric equations for the line t

Question

Find the tangent vector Tbar(t) and a set of parametric equations for the line tangent to r(t) at the point (2, 2squarerrot3, 2): r(t) = 4 cos t t + 4sin t j + 2 k Find r(T), N(t), ar, ax: r(t)=ti + 3squarerrott j @t (Apply the 'short cut' to find N(t)) Find the vectors T, N, & B = T Times B for r(t) = 3 cos t i + 3 sin j + tk @ t = phi/2 Find the length of the curve r(t) = (3cos t. 3 sin t, t) over the interval [0, phi]. Find the curvature K and the radius of curvature for the following: r(t) = 3 cos t i + 3 sin t j + t k y = 2x-3 Find T(t), N(t), aj, aN: r(t) = 2 cos 3t i+ 2 sin 3t j (Use your results from HW #29, section 12.4)

Explanation / Answer

Solution: (1)   given r(t) = 4cost i + 4sint j + 2k, P(2, 2sqrt3, 2)

T(t) = r'(t)/||r'(t)||
r'(t) = <-4sint, 4cost, 0>
||r'(t)|| = (16sin^2 t + 16cos^2 t) = 4

T(t) = <-4sint, 4cost, 0>/ 4

At that point t = pi/3
T(Pi/3) = <-23, 2, 0> / 4

For the tangent line add the vector for the point times an arbitrary scalar to the position vector for the point.
x = 2 - 23t / 4 = 2 - 3t /2
y = 23 + 2t/4 = 23 + t/2
z = 2

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