Question

. Find the intersection of the planes x − y + 5z = 9 and

x = 1 + s − t

y = 1 +2 s − t

z = 2 − s + t .

(a) Find the line ℓ1 perpendicular to the first of these planes and passing across the point (1, 2, 2).

(b) Find a line ℓ2 perpendicular to the second of these planes and passing across the point (1, 2, 2).

(c) Find the angle between ℓ1 and ℓ2.

(d) Represent both planes, both lines and the given point in the traditional Cartesian coordinate system.

Answer #1

Consider the following planes. 5x − 4y + z = 1, 4x + y − 5z =
5
(a) Find parametric equations for the line of intersection of
the planes. (Use the parameter t.) (x(t), y(t), z(t)) = (b)
Find the angle between the planes. (Round your answer to one
decimal place.) °

Find the intersection of two planes 3x + 2y − 5z = 3 and x − y −
2z = 4.

Find a plane containing the point (-6,-5,2) and the line of
intersection of the planes −7x−y−7z=−55 and x+2y−5z=−19

Consider the following planes.
x + y + z = 1, x + 3y + 3z = 1
(a) Find parametric equations for the line of intersection of
the planes. (Use the parameter t.)
(x(t), y(t), z(t)) =
(b) Find the angle between the planes. (Round your answer to one
decimal place.)
°

1) Find an equation of the plane. The plane through the point
(7, 0, 4)and perpendicular to the line x = 3t,y = 3 − t,z = 1 +
7t
2) Consider the following planes.x + y + z = 2, x + 6y + 6z =
2
(a) Find parametric equations for the line of intersection of
the planes. (Use the parameter t.) (x(t), y(t), z(t))
=
(b)Find the angle between the planes. (Round your answer to one
decimal...

Find a plane containing the point (2,8,6) and the line of
intersection of the planes − 8 x + 5 y + 7 z = 126 and 8 x + 4 y +
4 z = − 24.

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.

Find the line intersection and the angle between the planes
3x-2y+z=1 and 2x+y-3z=3.

Find an equation for each of the following planes. Use x, y and
z as the variables.
a) An equation of the plane passing through the points (1,−1,1),
(0,−2,−1) and (−4,0,6)
b) An equation of the plane consisting of all points that are
equidistant (equally far) from (−3,−5,−1) and (4,−1,−3)
c) An equation of the plane containing the line
x(t)= [0, -1, 1] + t[0, 4, -1] and is
perpendicular to the plane 3y − 4z = −7

Find an equation of the plane.
The plane that passes through the line of intersection of the
planes
x − z = 2 and y + 3z = 1
and is perpendicular to the plane
x + y − 3z = 3

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