MAT 21D Final: MAT 21D – Lecture 12 – Arc Length in Space

Determine whether the set of all third-degree polynomial functions as given below, with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails. ax^3 + bx^2 + cx + d, a notequalto 0 Verity W = {(x, y, 3x-8y): x and y are real numbers} is a subspace of V = R^2. Determine whether the set S = {(-5, 6), (3, 7)} is linearly independent or linearly dependent. Determine whether the set S = {(3, 1), (-1, 3)} spans R^2. Explain why S = {(6, -5), (12, -10)} is not a basis for R^2. Find the rank of the matrix [-4 3 0 -24 18 0 24 -18 0 11 -5 1]. Find a basis for the subspace of R^3 spanned by S = {(16, 4, 30), (8, 2, 15), (4, 1, 5)}. Find the length of the vector v = (2, 4, 2). Find the distance d between u = (3, 3, 6) and v = (-3, 3, 0). Find the angle theta between the vectors u = (0, 5, 0, 5) and v = (3, 2, 7, 3). Find the vector v with length 3 and the same direction as the vector u = (-5, 5, 1).

11 Find the point on the curve r(t) (12sin t)l (12cos tj 5t k at a distance of 13n units along the curve from the point (0, -12, 0) in the direction opposite to the direction of increasing arc length. 12 Compute T, N, and K at the point 1 of the twisted cubic with position 2' 3 vector r t i j at k. TT 13 Find T, N, Br K, and T for the space curve r() (cos3t)i (sin3 t j, o t 14 Given r t) et cos t)i (et sin t)j 2e k, write a in the form a aTT aNN atta 0 without finding Tand N. 15 Given r(t) (e t cos t)l (et sin tj 2k. Find r, T, N, and B at t 0. Then find the equations for the osculating, normal, and rectifying planes at t 0.

Find the length of the arc of the circular helix with vector equation r(t) = t, cos(pit), sin(pit) from the point (0, 1, 0) to the point (2, 1, 0). Find an equation for the NORMAL PLANE to the curve r(t) = t, cos(pit), sin(pit) when t = 1/4. what point in R3 corresponds to r (1/4)? Use a computer algebra system to produce a picture of the helix, plane and point you identified in part b).