Find the distance between the skew lines with parametric
equations x = 1 + t, y = 3 +
6t, z = 2t, and
x = 1 + 2s, y = 6 + 15s, z
= −2 + 6s.
Find the equation of the line that passes through the points on
the two lines where the shortest distance is measured.
3. (5 points) (a): Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.$$ x=e^{-t} \cos t, \quad y=e^{-t} \sin t, \quad z=e^{-t} ; \quad(1,0,1) $$(b): Find the unit tangent vector \(\mathbf{T}\), the principal unit normal \(\mathbf{N}\), and the curvature \(\kappa\) for the space curve,$$ \mathbf{r}(t)=<3 3="" 4="" sin="" cos="" t="">$$
Find the vector and parametric equations for the plane. The
plane that contains the lines r1(t) = <6, 8, 8,> + t<-2,
9, 6> and r2 = <6, 8, 8> + t<5, 1, 7>.
Find the parametric equations of the line of intersection of the
planes x − z = 1 and y + 2z = 3. (b) Find an equation of the plane
that contains the line of intersection above and it is
perpendicular to the plane x + y − 2z = 1.
Determine whether the following two lines are parallel,
intersecting, or skew. If they are skew, find the distance between
them
L1 : x = 3 + 2t, y = 4 − 3t, z = −1 − 4t and
L2 : 1 + 2s, y = 2 + s, z = 3 + 2s
Determine whether the following two lines are parallel,
intersecting, or skew. If they are skew, find the distance between
them L1 : x = 3 + 2t, y = 4 − 3t, z = −1 − 4t and L2 : 1 + 2s, y =
2 + s, z = 3 + 2s
1. Given parametric equations below, find the values of t where
the the parametric curve has a horizontal and vertical
tangents.
a) x=t^2 - t, y= t^2 + t
b) x= e^(t/10)cos(t), y= e^(t/10)sin(t)
2. Find the arc length of the graph of the parametric equations
on the given intervals.
a) x= 4t+2, y = 1-3t , −1 ≤ t ≤ 1
b) x= e^(t/10)cos(t), y= e^(t/10)sin(t), 0 ≤ t ≤ 2π
Find the equations of the tangent lines to the circle x^2 + y^2
= 9 which pass through the point (−7, 2).
(Ans.: Using decimal approximations, the lines are approximately
y = −0.847493x − 3.932456 and 0.147493x + 3.032456.)