Question

Consider the lines ℓ1 = { (−1 + 2t, t, α − t) |t ∈ R} and ℓ2 = { (2s, α + s, 3s) | s ∈ R }.

(a) There is only one real value for the unknown α such that the lines ℓ1 and ℓ2 intersects each other at a point P ∈ R^3 . Calculate that value for α and determine the point P.

(b) If α = 0, is there a plane π ⊂ R^3 containing both lines ℓ1 and ℓ2? Justify your answer.

Answer #1

Determine whether the lines x = [3−t, 2+t, 8+2t] and x = [2+2s,
−2+3s, −2+ 8s] intersect and if so, find the point of
intersection.

Consider the lines in space whose parametric equations are as
follows
line #1 x=2+3t, y=3-t, z=2t
line #2 x=6-4s, y=2+s, z=s-1
a Find the point where the lines intersect.
b Compute the angle formed between the two lines.
c Compute the equation for the plane that contains these two
lines

Determine whether the line (x,y,z) = r(t) = (1-t, 4-5t,
2t+5)
a. Intersects the xy plane
b. Intersects with the z-axis

a) Determine: L{t^3 e^2t+e^2t sin( 5t)} and b) Find L^(-1)
{(3s+2)/(s^2+2s+10)}

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.

1. Consider the plane 4x+y-2z=4 and
the line r(t) = < t, -2t,
-tt >.
a. find the unit normal vector N of the plane.
b. as a function of t find the distance between
r(t) and the plane.
2. Consider a fruit fly flying a room with velocity
v(t) = < -sin(t), cos(t), 1 >
a. if the z = 1 + 2(pi) is the room's ceiling, where
will the fly hit the ceiling?
b. if the temperature in...

Consider plane P: 4x -y + 2z = 8, line: <x, y, z> =
<1+t, -1+2t, 3t>, and point Q(2,-1,3)
b) Find the perpendicular distance between point Q and plane
P

Find the curvature of ~r(t) = (t3 −5)~ i + (t4 + 2)~ j + (2t +
3)~ k at the point P(−6,3,1)?

Determine how the following lines interact.
(x, y, z) = (-2, 1, 3) + t(1, -1, 5) ; (x, y, z) =
(-3, 0, 2) + s(-1, 2, -3)
(x, y, z) = (1, 2, 0) + t(1, 1, -1) ; (x, y, z) =
(3, 4, -1) + s(2, 2, -2)
x = 2 + t, y = -1 + 2t, z = -1 – t ; x = -1 - 2s,
y = -1 -1s, z = 1...

Determine whether the lines
L1:→r(t)=〈−2,−1,3〉t+〈−5,−3,−1〉 and
L2:→p(s)=〈4,2,−6〉s+〈4,−1,0〉
intersect. If they do, find the point of intersection.

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