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Lines and Planes
1.
2.
•
3.
4.
•
Find a set of parametric equations and also a set of symmetric equations of the
line through the given point parallel to the indicated line or vector. For each line,
express the direction numbers as integers:
(a)
(0,0,0)
v = (1,2,3)
(b)
(-2,0,3)
v = 2i+4j-2k
(c)
(1,0,1)
x=3+3t, y=5-2t, z= -7+1
(d)
(-3,5,4)
(x-l)/3 = (y+I)/-2 = z-3
Find a set of parametric equations and also a set of symmetric equations of the
line through the two points (express the direction numbers as integers):
(a)
(5,-3,-2), (-2/3,2/3, I)
(b)
(1,0,1), (1,3,-2)
Find a set of parametric equations of the line described:
(a)
The line passes through (2,3,4) and is parallel to the xz-plane and the yz­
plane.
(b)
The line passes through (2,3,4) and is perpendicular to the plane given by
3x+2y-z = 6.
Determine whether the following pairs oflines intersect, and if so, find the point
of intersection and the cosine ofthe acute angle between the planes:
(a)
Line #1 is: x=4t+2, y = 3, z = -t+1
Line #2 is: x=2s+2, y=2s+3, z = s+1
(b)
Line #1 is: x/3=(y-2)/-I=z+1.
Line #2 is: (x-I)/4 = y+2 = (Z+3)/-3
..
Page 53
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•
5.
Find an equation of the plane passing through the point that is perpendicular to the
given vector or line:
(a)
(2,1,2)
n= i
(b)
(3,2,2)
D
(c)
(0,0,6)
x=1-t, Y = 2+t, z = 4-21.
= 2i+3j-k
6.
Find an equation of the plane passing through (0,0,0), (1,2,3) and (-2,3,3).
7.
Find an equation of the plane containing the lines given by:
Line #1: (x-l)/-2 = y-4 =z and Line #2: (x-2)/-3 = (y-l)/4 = (z-2)/-1.
8.
Find an equation of the plane passing through the points (2,2,1) and (-1,1,-1) that
is perpendicular to the plane 2x-3y+z = 3.
9.
Detennine the angle of intersection ofthe following pairs of planes:
10.
11.
(a)
5x-3y+z = 4 and x+4y+7z = l.
(b)
x-3y+6z = 4 and 5x+y-z = 4.
(c)
x-5y-z = 1 and 5x-25y-5z = -3
Mark the intercepts and make a rough sketch ofthe graph of each of the following
planes:
(a)
4x+2y+6z = 12
(b)
2x-y+3z=4
(c)
y+z= 5
Find the intersection point, ifany, ofthe point and the line. Also determine
whether the line lies in the plane:
Plane: 2x-2y+z = 12
line: x-l/2 = (y+3/2)/-1
•
= (z+1)/2
12.
Find the distance between the point (1,2,3) and the plane 2x+3y+z = 4.
13.
Find the distance between the planes x-3y+4z = 10 and x-3y+4z = 6.
Page 54
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