Question

- 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*.

Answer #1

Two lines are perpendicular if their sum of product of direction cosines is zero.

please ask for any doubt and rate if u satisfied

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 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

1. Let T(x, y, z) = (x + z, y − 2x, −z + 2y) and S(x, y, z) =
(2y − z, x − z, y + 3x). Use matrices to find the composition S ◦
T.
2. Find an equation of the tangent plane to the graph of x 2 − y
2 − 3z 2 = 5 at (6, 2, 3).
3. Find the critical points of f(x, y) = (x 2 + y 2 )e −y...

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...

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)....

Find parametric equations of the line that passes through (1, 2,
3) and is parallel to the plane
2 x − y + z = 3 and is perpendicular to the line with parametric
equations:
x=5-t, y=3+2t , z=4-t

Find parametric equations of the line that passes through (1, 2,
3) and is parallel to the plane
2 x − y + z = 3 and is perpendicular to the line with parametric
equations:
x=5-t, y=3+2t, z=4-t

Find the point(s) of intersection, if any, of the line
x-2/1 = y+1/-2 = z+3/-5 and the plane 3x + 19y - 7a - 8 =0

Find parametric equations for the line through (1, 1, 6) that is
perpendicular to the plane x − y + 3z = 8. (Use the parameter t.)
(x(t), y(t), z(t)) = 1+t, 1-t, 6+3t Correct: Your answer is
correct.
(b) In what points does this line intersect the coordinate
planes? xy-plane (x, y, z) = yz-plane (x, y, z) = xz-plane (x, y,
z) =

1. Solve the following system of equations by the elimination
method:
2x+y-z=7
x+2y+z=8
x-2y+3z=2
2. Solve the following system of equations by using row
operations on a matrix:
2x+y-z=7
x+2y+z=8
x-2y+3z=2

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