Samantha Rodriguez Brewer MAT 267 ONLINE B Fall 2020
Assignment Section 10.1 due 10/15/2020 at 11:59pm MST
1. (1 point) What are the projections of the point
(−5,−9,−6)on the coordinate planes?
On the xy-plane: ( , , )
On the yz-plane: ( , , )
On the xz-plane: ( , , )
Solution:
SOLUTION:
The projection of (−5,−9,−6)onto the xy-plane is
(−5,−9,0).
The projection of (−5,−9,−6)onto the yz-plane is
(0,−9,−6).
The projection of (−5,−9,−6)onto the xz-plane is
(−5,0,−6).
Correct Answers:
•-5
•-9
•0
•0
•-9
•-6
•-5
•0
•-6
2. (1 point) Determine whether the three points P=
(9,−8,−8),Q= (8,−10,−11),R= (7,−11,−14)are colin-
ear by computing the distances between pairs of points.
Distance from Pto Q:
Distance from Qto R:
Distance from Pto R:
Are the three points colinear (y/n)?
Solution:
SOLUTION:
Distance from Pto Q:
p(8−9)2+ (−10 +8)2+ (−11 +8)2=1√14
Distance from Qto R:
p(7−8)2+ (−11 +10)2+ (−14 +11)2=√11
Distance from Pto R:
p(7−9)2+ (−11 +8)2+ (−14 +8)2=√49
In order for the points to lie on a straight line, the sum of the
two shortest distances must equal the longest distance. Since
1√14 +√11 6=√49, the three points do not lie on a straigh
line.
Correct Answers:
•3.74165738677394
•3.3166247903554
•7
•N
3. (1 point) What is the distance from the point (8,6,−5)to
the xz-plane?
Distance =
Solution:
SOLUTION:
The distance from the point to the xz-plane is the absolute
value of the y-coordinate of the point. Thus the distance is 6.
Correct Answers:
•6
4. (1 point) What do the following equations represent in R3?
Match the two sets of letters:
a. a vertical plane
b. a horizontal plane
c. a plane which is neither vertical nor horizontal
A. −1x+6y=9
B. x=−3
C. y=−2
D. z=−6
Correct Answers:
•A
•A
•A
•B
5. (1 point) Find the equation of the sphere centered at
(0,5,2)with radius 4.
= 0.
Give an equation which describes the intersection of this
sphere with the plane z=3.
= 0.
Solution:
SOLUTION:
An equation of the sphere with center (0,5,2)and radius 4 is
(x−0)2+ (y−5)2+ (z−2)2=42or
(x−0)2+ (y−5)2+ (z−2)2−42=0 .
The intersection of this sphere with the plane z=3 is the set
of points on the sphere whose z-coordinate is z=3. Putting
z=3 into the equation yields (x−0)2+(y−5)2+1−42=0 or
(x−0)2+ (y−5)2−15 =0.
This is a circle in the plane z=3 with center (0,5,3)and radius
√15.
Correct Answers:
1
•(x - 0)**2 + (y - 5)**2 + (z - 2)**2 - 4**2
•(x - 0)**2 + (y - 5)**2 + 1 - 4**2
6. (1 point) Find the equation of the sphere if one of its di-
ameters has endpoints (−3,3,3)and (−2,5,6).
= 0.
Solution:
SOLUTION:
The center of the sphere is the midpoint of the diameter:
−3−2
2,3+5
2,3+6
2= (−2.5,4,4.5).
The radius is half the diameter, so
r=1
2p(−2+3)2+ (5−3)2+ (6−3)2=1
2√14.
Therefore an equation of the sphere is
(x+2.5)2+ (y−4)2+ (z−4.5)2−14
4=0
Correct Answers:
•(x - -2.5)**2 + (y - 4)**2 + (z - 4.5)**2 - 1.87082869338697**2
7. (1 point) Find an equation of the sphere that passes
through the origin and whose center is (3,−1,6).
= 0
Note that you must put everything on the left hand side of the
equation and that we desire the coefficients of the quadratic
terms to be 1.
Solution:
SOLUTION:
The radius of the sphere is the distance from the center to the
origin: r=√32−12+62=√46. Therefore the equation of the
sphere is
(x−3)2+ (y+1)2+ (z−6)2−46 =0
Correct Answers:
•xˆ2 + yˆ2 + zˆ2 + (-6*x + 2*y + -12*z)
8. (1 point) Find an equation of the largest sphere with center
(8,2,7)that is contained completely in the first octant.
= 0
Note that you must move everything to the left hand side of the
equation that we desire the coefficients of the quadratic terms to
be 1.
Solution:
SOLUTION:
The largest sphere contained in the first octant must have a
radius equal to the minimum distance from the center (8,2,7)
to any of the three coordinate planes. The shortest distance is 2,
thus an equation of the sphere is
(x−8)2+ (y−2)2+ (z−7)2−22=0
Correct Answers:
•xˆ2 + yˆ2 + zˆ2 - 2*(8*x + 2*y + 7*z) - 2ˆ2 + (8ˆ2 + 2ˆ2 +
7ˆ2)
9. (1 point) Find the center and radius of the sphere
x2−2x+y2−18y+z2−0z=−18
Center: ( , , )
Radius:
Solution:
SOLUTION:
Completing the squares in the equation gives
(x2−2x+1)+(y2−18y+81)+(z2−0z+0) = −18+1+81 +
0
⇒(x−1)2+ (y−9)2+ (z−0)2=64,
which we recognize as an equation of a sphere with center
(1,9,0)and radius 8.
Correct Answers:
•1
•9
•0
•8
10. (1 point) Write down an (in)equality which describes the
solid ball of radius 8 centered at (3,−1,−8).It should have a
form like x2+y2+ (z−2)2−4>=0, where you use one of the
following symbols ≤, <, =,≥, >.
The first blank is for the algebraic expression; the drop-down
list gives the (in)equatilty.
? 0.
Solution:
SOLUTION:
The solid ball consists of all the points on or inside the sphere
with radius 8 and center at (3,−1,−8). This set of points is de-
scribed by the inequality (x−3)2+ (y+1)2+ (z+8)2≤64, or,
equivalently, (x−3)2+ (y+1)2+ (z+8)2−64 ≤0.
Correct Answers:
•(x - 3)**2 + (y - -1)**2 + (z - -8)**2 - 8**2
•<=
11. (1 point)
You are given the following points: A= (−4,17,19),
B= (17,0,1),C= (6,17,−20).
Which point is closest to the yz-plane? [?/A/B/C]
What is the distance from the yz-plane to this point?
Which point is farthest from the xy-plane? [?/A/B/C]
What is the distance from the xy-plane to this point?
Which point lies on the xz-plane? [?/A/B/C]
Solution:
SOLUTION
The distance from a point to the yz-plane is the absolute value
of the x-coordinate.
The point A(−4,17,19)has the xcoordinate with the smallest
absolute value, so Ais the point closest to the yz- plane.
The distance from the yz-plane to Ais given by the absolute
value of the x-coordinate, i.e. |−4|=4.
The distance from a point to the xy-plane is the absolute value
of the z-coordinate.
The point C(6,17,−20)has the zcoordinate with the largest
absolute value, so Cis the point farthest from the xy- plane.
2
The distance from the xy-plane to Cis given by the absolute
value of the z-coordinate, i.e. |−20|=20.
A point lies on the xz-plane if its y-coordinate is zero. Thus
B(17,0,1)lies on the xz-plane.
Correct Answers:
•A
•4
•C
•20
•B
12. (1 point) Find the distance from (−7,2,−12)to each of
the following:
1. The xy-plane.
Answer:
2. The yz-plane.
Answer:
3. The xz-plane.
Answer:
4. The x-axis.
Answer:
5. The y-axis.
Answer:
6. The z-axis.
Answer:
Solution:
SOLUTION
1. The distance from a point to the xy-plane is the abso-
lute value of the z-coordinate of the point. Thus, the distance is
|−12|=12.
2. The distance from a point to the yz-plane is the abso-
lute value of the x-coordinate of the point. Thus, the distance
is |−7|=7.
3. The distance from a point to the xz-plane is the abso-
lute value of the y-coordinate of the point. Thus, the distance
is |2|=2.
4. The point on the x-axis closest to (−7,2,−12)is the point
(−7,0,0), (Approach the x-axis perpendicularly.)
The distance from (−7,2,−12)to the x-axis is the distance be-
tween these two points:
p(−7+7)2+ (2−0)2+ (−12 −0)2=p(2)2+ (−12)2=
√148
5. The point on the y-axis closest to (−7,2,−12)is the point
(0,2,0), (Approach the y-axis perpendicularly.)
The distance from (−7,2,−12)to the y-axis is the distance be-
tween these two points:
p(−7−0)2+ (2−2)2+ (−12 −0)2=p(−7)2+ (−12)2=
√193
6. The point on the z-axis closest to (−7,2,−12)is the point
(0,0,−12), (Approach the z-axis perpendicularly.)
The distance from (−7,2,−12)to the z-axis is the distance be-
tween these two points:
p(−7−0)2+ (2−0)2+ (−12 +12)2=p(−7)2+ (2)2=
√53
Correct Answers:
•|-12|
•|-7|
•|2|
•sqrt(2ˆ2+(-12)ˆ2)
•sqrt((-7)ˆ2+(-12)ˆ2)
•sqrt((-7)ˆ2+2ˆ2)
13. (1 point) Match the equations of the plane with one of
the graphs below.
A B C
D E F
1. x−z=2
2. x+y=−2
3. x+z=2
4. y−x=2
Note: You can click on the graphs to enlarge the images.
Solution:
SOLUTION
1. The plane x−z=2 is a plane parallel to the y-axis, that
intersects the x-axis at the point (2,0,0)and the z-axis at the
point (0,0,−2). Thus the equation matches the graph E.
2. The plane x+y=−2 is a vertical plane that intersects the
xy-plane in the line y=−2−x. Thus the equation matches the
graph C.
3. The plane x+z=2 is a plane parallel to the y-axis, that
intersects the x-axis at the point (2,0,0)and the z-axis at the
point (0,0,2). Thus the equation matches the graph D.
4. The plane y−x=2 is a vertical plane that intersects the
xy-plane in the line y=2+x. Thus the equation matches the
graph B.
Correct Answers:
•E
•C
•D
•B
3
14. (1 point) Match the equations of the spheres with one of
the graphs below.
A B C
D E F
1. x2−4x+y2+z2=−15
4
2. x2−4x+y2−4y+z2−2z=−35
4
3. x2+y2+ (z+1)2=9
4
4. x2−2x+y2+2y+z2−2z=−2
Note: You can click on the graphs to enlarge the images.
Solution:
SOLUTION
1. Completing the squares, yields (x−2)2+y2+z2=1
4.
Thus the sphere is centered at (2,0,0)and has radius 1
2. Its
graph matches F.
2. Completing the squares, yields (x−2)2+ (y−2)2+ (z−
1)2=1
4. Thus the sphere is centered at (2,2,1)and has radius
1
2. Its graph matches D.
3. The sphere is centered at (0,0,−1)and has radius 3
2. Thus
it matches C.
4. Completing the squares, yields (x−1)2+ (y+1)2+ (z−
1)2=1. Thus the sphere is centered at (1,−1,1)and has radius
1. Its graph matches E.
Correct Answers:
•F
•D
•C
•E
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