Question

(a) Find the distance between the skew lines l1 and l2 given with the vector equations l1 : r1(t) = (1+t)i+ (1+6t)j+ (2t)k; l2 : r2(s) = (1+2s)i+ (5+15s)j+ (−2+6s)k.

(b) Determine if the plane given by the Cartesian equation −x + 2z = 0 and the line given by the parametric equations x = 5 + 8t, y = 2 − t, z = 10 + 4t are orthogonal, parallel, or neither.

Answer #2

answered by: anonymous

Find the exact distance between the two skew lines given by r(t)
=< 2t + 1, 3t +1, 4t +1> and r(t) = <2t + 3, -t + 2, t +
3> using the vector formulas involving dot products or cross
products.

Find vector and parametric equations for:
a) the line that passes through the point P(9,-9,6) parallel to
the vector u = <3,4,-2>
b) the line passing through the point P(6,-2,6) parallel to the
line x=2t, y = 2 - 3t, z = 3 +6t
c) the line passing through the point P(5, -2,1) parallel to the
line x = 3 - t, y = -2 +4t, z = 4 + 8t

At what point do the curves r1 =〈 t
, 1 − t , 3 + t2 〉 and r2 =
〈 3 − s , s − 2 , s2 〉 intersect? Find the angle of
intersection.
Determine whether the lines L1 :
r1 = 〈 5 − 12t , 3 + 9t ,1 − 3t 〉 and
L2 : r2 = 〈 3 + 8s , −6s , 7
+ 2s 〉are parallel, skew, or intersecting. Explain. If...

Given the following pairs of lines, determine whether they are
parallel, intersecting or skew, if they intersect, find the
intersecting and the plane containing them.
1) L1: (x-1)/1=(y-2)/1=(z-3)/-2
L2:(x-1)/1=(y-3)/0=(z-2)/-1
2) L1: x=t, y=-t,z=-1 L2: x=s, y=s,
z=5
2) L1: (3+2x)/0=(-3+2y)/1=(6-3z)/2 L2:
x=5/2, y=(3/2)-3t, z=2+4t

Find the shortest distance between line L1: x = 1 + 2t, y = 3
- 4t, z = 2 + t and L2: the intersection of the planes x + y + z =1
and 2x + y - 3z = 10

Find a vector equation and parametric equations for the line.
(Use the parameter t.)
The line through the point
(0, 12, −8)
and parallel to the line
x = −1 + 3t,
y = 6 − 4t, z
= 3 + 6t

20. Find the unit tangent vector T(t) and then use it to find a
set of parametric equations for the line tangent to the space curve
given below at the given point.
r(t)= -5t i+ 2t^2 j+3tk, t=5

. Find the flux of the vector field F~ (x, y, z) =
<y,-x,z> over a surface with downward orientation, whose
parametric equation is given by r(s, t) = <2s, 2t, 5 − s 2 − t 2
> with s^2 + t^2 ≤ 1

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

Find the distance between the given skew lines x=5-t, y=4+5t,
z=1-3t and x=t, y=9, z=5t
KINDLY INCLUDE THE COMPLETE SOLUTION

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