Given the vector valued function R(t)=<cos(t), sin(t), ln(cos(t)> with t being bounded from -pi/2 to positive without being equal to pi/2 or -pi/2

binormal-vectors

#1

Given the vector valued function R(t)=<cos(t), sin(t), ln(cos(t)> with t being bounded from -pi/2 to positive without being equal to pi/2 or -pi/2, find the Tangent, Normal and Binormal vectors at the point (1,0,0)

Answer:

r(t) = <cost, sint, ln(cost)>
r’(t) = <-sint, cost, -tant>
||r’(t)|| = sqrt(sin^2t + cos^2t + tan^2t) = sqrt(1 + tan^2t) = sect
T(t) = <-sint/sect, cost/sect, -tant/sect> = <-sintcost, cos2t, -sint>
T’(t) = <sin^2t – cos^2t, -2costsint, -cost>

Note: at <1, 0 , 0>, t = 0
T(0) = <0, 1, 0>
T’(0) = < -1, 0, -1> ; ||T’(0)|| = sqrt(2)
N(0) = <-1/sqrt(2), 0, -1/sqrt(2)>
B(0) = T(0) X N(0) = <-1/sqrt(2), 0, 1/sqrt(2)>