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=
Suppose r(t)=cos(πt)i+sin(πt)j+2tk represents the position of a
particle on a helix, where z is the height of the particle. (a)
What is t when the particle has height 8? (b) What is the velocity
of the particle when its height is 8? (c) When the particle has
height 8, it leaves the helix and moves along the tangent line at
the constant velocity found in part (b). Find a vector parametric
equation for the position of the particle (in terms...
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