Question

3. Consider the following two lines:

x = c + t, y = 1 + t, z = 5 + t and x = t, y = 1 - t, z = 3 + t.

Is there a value c that makes the two lines intersect? If so, find it. Otherwise, give a reason.

4. A particle starts at the origin and moves along the shortest path to the line determined by the two points P =(1,2,3) and Q =(3,-2,-1).

(a) Find the location where the particle hits the line.

(b) If the particle moves with constant speed and reaches the line in 2 seconds. Parameterize its motion of the first 2 seconds.

5. With each stated property, give an example of the 2-space vector field F(x,y).

(a) F has a constant direction, but ||F|| increases when moving away from the origin.

(b) F is perpendicular to the vector field <x,2> and ||F||=5 everywhere.

Answer #1

Show that the two lines with equations (x, y, z) = (-1, 3,
-4) + t(1, -1, 2) and (x, y, z) = (5, -3, 2) + s(-2, 2,
2) are perpendicular. Determine how the two lines
interact.
Find the point of intersection of the line (x, y, z) = (1,
-2, 1) + t(4, -3, -2) and the plane x – 2y + 3z =
-8.

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

The planes x + 3 y − 2 z = 1 and 2 x − y + 3 z = 4 intersect in
a line. A direction vector for this line is given by:

1. Determine whether the lines are parallel, perpendicular or
neither. (x-1)/2 = (y+2)/5 = (z-3)/4 and (x-2)/4 = (y-1)/3 =
(z-2)/6
2. A) Find the line intersection of vector planes given by the
equations -2x+3y-z+4=0 and 3x-2y+z=-2
B) Given U = <2, -3, 4> and V= <-1, 3, -2> Find a. U
. V b. U x V

Consider the vector field F = <2 x
y^3 , 3 x^2
y^2+sin y>. Compute
the line integral of this vector field along the quarter-circle,
center at the origin, above the x axis, going from the point (1 ,
0) to the point (0 , 1). HINT: Is there a potential?

The force exerted by an electric charge at the origin on a
charged particle at a point (x, y, z) with position vector r = x,
y, z is F(r) = Kr/|r|3 where K is a constant. Find the work done as
the particle moves along a straight line from (3, 0, 0) to (3, 2,
5).

The force exerted by an electric charge at the origin on a
charged particle at a point (x, y, z)
with position vector r = <x,
y, z> is
F(r) =
Kr/
|r|3 where K is a constant.
Find the work done as the particle moves along a straight line from
(2, 0, 0) to (2, 4, 5)

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...

1)T F: All (x, y, z) ∈ R 3 with x = y + z is a subspace of R 3
9
2) T F: All (x, y, z) ∈ R 3 with x + z = 2018 is a subspace of R
3
3) T F: All 2 × 2 symmetric matrices is a subspace of M22. (Here
M22 is the vector space of all 2 × 2 matrices.)
4) T F: All polynomials of degree exactly 3 is...

particle of mass m in R3 has position function r(t)
=<x(t),y(t),z(t)>. Given that the tangent vector r0(t) has a
constant length of 5, please prove that at all t values, the force
F(t) acting on the particle is orthogonal to the tangent vector

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