Equation Worksheet With Answer Key
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2
3WORKSHEET 12
1. Instructions
I will collect any work you want to turn in on March 7th. You are not expected to
turn in all the problems, but you should choose the problems you think will most benefit
you and work on those.
2. Lines and Planes
Problem 1. Convert the following vector equations for lines to parametric equations for
lines:
(1) r = (5
t) + 6 + t ;
(2) r = 5 + 6t .
Answer.
(1) x = 5
t, y = 6 and z = t.
Problem 2. Find a vector equation for the line that passes through the point (5, 1, 3) and
is parallel to the vector + 4
2 . Find two other points on this line.
Answer. r = (5 + t) i + (1 + 4t) j + (3
2t) k.
Problem 3. Find parametric and vector equations of the line passing through the point
A = (2, 4, 3) and B = (3, 1, 1). Where does this line intersect the xy-plane?
Answer. Vector equation: r = (2 + t) i + (4
5t) j + ( 3 + 4t) k. Intersects the xy-plane at
the point (11/4, 1/4, 0).
Problem 4. Show that the lines L
and L
with parametric equations
1
2
L
:
x = 1 + t
y =
2 + 3t
z = 4
t
1
L
:
x = 2s
y = 3 + s
z =
3 + 4s
2
are skew lines, that is they neither intersect, nor are they parallel.
Problem 5. Find the angle between the planes x + y + z = 1 and x
2y + 3z = 1. The
two planes intersect in a line. Find a vector equation of that line.
Problem 6. Find an equation of the plane
(1) through the point (6, 3, 2) and perpendicular to the vector
2, 1, 5 ;
(2) through the point (1, 1, 1) and with normal vector +
;
(3) through the origin and parallel to the plane 2x
y + 3z = 1;
(4) that contains the line x = 3 + 2t, y = t and z = 8
t and is parallel to the plane
through 2x + 4y + 8z = 17;
(5) that passes through the point ( 1, 2, 1) and contains the line of intersection of the
planes x + y + z = 2 and 2x
y + 3z = 1.
Answer.
(1)
2x + y + 5z = 1;
(2) x + y
z =
1;
(3) 2x
y + 3z = 0;
Date: March 5th-, 2012.
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