Question

Let T be the plane 2x+y = −4. Find the shortest distance d from
the point P_{0}=(−1, −5, −1) to T, and the point Q in T
that is closest to P_{0}. Use the square root symbol '√'
where needed to give an exact value for your answer.

Answer #1

a) Let T be the plane 3x-y-2z=9. Find the shortest distance d
from the point P0 = (5, 2, 1) to T, and the point Q in T that is
closest to P0. Use the square root symbol where needed.
d=
Q= ( , , )
b) Find all values of X so that the triangle with vertices A =
(X, 4, 2), B = (3, 2, 0) and C = (2, 0 , -2) has area (5/2).

Let L1 be the line passing through the point
P1(?5, ?4, 5) with direction vector
d=[?1, 1, 3]T, and let
L2 be the line passing through the point
P2(4, 1, ?4) with the same direction
vector.
Find the shortest distance d between these two lines, and
find a point Q1 on L1 and a
point Q2 on L2 so that
d(Q1,Q2) = d. Use
the square root symbol '?' where needed to give an exact value for
your answer.
d=?...

Let L1 be the line passing through the point
P1=(−5, −2, −5) with direction vector
→d=[2, −3, −2]T, and let L2
be the line passing through the point P2=(4,
−1, −5) with the same direction vector.
Find the shortest distance d between these two lines, and
find a point Q1 on L1 and a
point Q2 on L2 so that
d(Q1,Q2) = d. Use
the square root symbol '√' where needed to give an exact value for
your answer.

Let L1 be the line passing through the point
P1(3, 5, ?5) with direction vector
d=[?1, 2, 0]T, and let
L2 be the line passing through the point
P2(?3, ?4, ?3) with the same direction
vector.
Find the shortest distance d between these two lines, and
find a point Q1 on L1 and a
point Q2 on L2 so that
d(Q1,Q2) = d. Use
the square root symbol '?' where needed to give an exact value for
your answer.

Let L1 be the line passing through the point
P1(?5, ?3, ?2) with direction vector
d=[0, ?3, ?2]T, and let
L2 be the line passing through the point
P2(?2, 3, ?3) with the same direction
vector.
Find the shortest distance d between these two lines, and
find a point Q1 on L1 and a
point Q2 on L2 so that
d(Q1,Q2) = d. Use
the square root symbol '?' where needed to give an exact value for
your answer.
d...

Find the shortest distance from the point P = (−1, 2, 3) to the
line of inter- section of the planes x + 2y − 3z = 4 and 2x − y +
2z = 5.

Let P be the plane given by the equation 2x + y − 3z = 2. The
point Q(1, 2, 3) is not on the plane P, the point R is on the plane
P, and the line L1 through Q and R is orthogonal to the plane P.
Let W be another point (1, 1, 3). Find parametric equations of the
line L2 that passes through points W and R.

Find the shortest distance between the point (1, 4, 2) and the
paraboloid described by z = x^2 + y^2 . Figure out a way to do this
without having to deal with square roots.

Find an equation for the plane that contains the
line v=(4,−4,−2)+t(3,−3,1)
and is perpendicular to the plane 2x+y+4z+1=0
(Use symbolic notation and fractions where needed. Please note
that the solution expects you to solve for zz. You may need to
scale your answer suitably. )
z = .

Find the minimum distance from the point (1,-6,3) to the plane x
− y + z = 7. (Hint: To simplify the computations, minimize
the square of the distance.)

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