Question

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).

Answer #1

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, −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.

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

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 f(x,y) = 9y^2 −(3x^2)y denote the temperature at the point
(x,y) in the plane, and let C(t) = (t^2, 3t) be the path of a
crawling ant in the plane. Find how fast the temperature of the ant
is changing at time t = 2.
At time t = 2 the ant is at the point C(2) = (4, 6). Which
direction should the ant crawl to warm up as quickly as possible
(in the near term)? Please a...

Find equations of the tangent plane and normal line to the
surface x=2y^2+2z^2−159x at the point (1, -4, 8).
Tangent Plane: (make the coefficient of x equal to 1).
=0.
Normal line: 〈1,〈1, , 〉〉
+t〈1,+t〈1, ,

Consider the plane x + 2y + z = 2 and the point P = (2,0,4) A)
Set up an equation to measure the distance d from P to an arbitrary
point (x,y,z) on the plane B) Find the pointe on the plane that is
closest to P C) What is the shortest distance between point P and
the plane?

Problem: Let y=f(x)be a differentiable function
and let P(x0,y0)be a point that is not on the graph of function.
Find a point Q on the graph of the function which is at a
minimum distance from P.
Complete the following steps. Let Q(x,y)be a point on the graph
of the function
Let D be the square of the distance PQ¯. Find an expression for
D, in terms of x.
Differentiate D with respect to x and show that
f′(x)=−x−x0f(x)−y0
The...

Find the point of intersection of the line
x(t) = (0, 1, 3) + (–2, – 1, 2)t
with the plane 4x + 5y – 4z = 9. And
Find the distance from the point (2, 3, 1) to the plane 3x
– 2y + z = 9

consider the plane x+2y+z=2 and the point P= (2,0,4)
a.) set up an equation to measure the distance d from P on an
arbitrary point (x,y,z) on the plane
b.) Find the point on the plane that is closest to P. hint it
may be easier to minimize d^2 (instead of d)
c.) What is the shortest distance between point P and the
plane

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