Consider the helix
r(t)=(cos(2t),sin(2t),−3t)r(t)=(cos(2t),sin(2t),−3t).
Compute, at t=π/6
A. The unit tangent vector T=T= ( , , )
B. The unit normal vector N=N= ( , , )
C. The unit binormal vector B=B= ( , , )
D. The curvature κ=κ=
Find the point of intersection of the tangent lines to the curve
r(t) = 5 sin(πt), 2 sin(πt), 6 cos(πt) at the points where t = 0
and t = 0.5. (x, y, z) =
Let F3={cos(t),sin(t),cos(3t),sin(3t)} and
T3={cos3(t),cos2(t)sin(t),cos(t)sin2(t),sin3(t)}. Use the power
reduction formulas and the triple angle identities to show the
following:
Show T3⊆Span(F3).
Show F3⊆Span(T3).
Given r(t) = <2 cos(t), 2 sin(t), 2t>. • What is the arc
length of r(t) from t = 0 to t = 5. SET UP integral but DO NOT
evaluate • What is the curvature κ(t)?
If u(t) = < sin(8t), cos(4t), t > and v(t) = < t,
cos(4t), sin(8t) >, use the formula below to find the given
derivative.
d/(dt)[u(t)* v(t)] =
u'(t)* v(t) +
u(t)* v'(t)
d/(dt)[u(t) x v(t)] =
<.______ , _________ , _______>
If u(t) = < sin(5t),
cos(5t), t > and
v(t) = < t, cos(5t),
sin(5t) >, use the formula below to find the given
derivative.
d/dt[ u(t) * v(t)] = u'(t) * v(t) + u(t)* v'(t)
d/dt [ u(t) x v(t)] = ?
Let L be the line parametrically by~r(t) = [1 + 2t,4 +t,2 + 3t]
and M be the line through the points P= (−5,2,−3) and
Q=(1,2,−6).
a) The lines L and M intersect; find the point of
intersection.
b) How many planes contain both lines?
c) Give a parametric equation for a plane Π that contains both
lines