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. Given the point
P
(

3
,
2
,
4)
,
a) what is its distance to xyplane? xzplane? yzplane? Write the equation
of the sphere centered at P that touches yzplane.
Answer.
The distance to xy plane is 4; The distance to xz plane is 2; The
distance to yz plane is 3; The equation of the sphere
(
x
+ 3)
2
+ (
y

2)
2
+ (
z

4)
2
= 3
2
.
b) Write the projections of
P
on xz, xy, and yzplanes.
Answer
. It is (

3
,
0
,
4) on xz plane; (

3
,
2
,
0) on
xy
plane, (0
,
2
,
4) on
yz
plane.
2.
Show that the equation represents a sphere, ﬁnd its radius and the center
coordinates:
x
2
+
y
2
+
z
2

4
x
+ 2
y

6
z
= 0
.
Answer.
Completing squares:
x
2
+
y
2
+
z
2

4
x
+ 2
x

6
x
=
x
2

4
x
+ 2
2
+
y
2
+ 2
y
+ 1
+
z
2

6
z
+ 3
2

14
= 0
or
(
x

2)
2
+ (
y
+ 1)
2
+ (
z

3)
2
= 14
.
Center at (2,1,3) and radius
√
14
.
3.
Use vectors to prove that the segment joining the midpoints of two sides
of a triangle is parallel to the third side and half its length.
Answer.
If A,B,C are vertices of the triangle, then
A
~
B

A
~
C
=
C
~
B
and the
vector joining the midpoints is
~a
=
1
2
A
~
B

1
2
A
~
C
=
1
2
±
A
~
B

A
~
C
²
=
1
2
C
~
B.
Therefore
~a
is parallel to
C
~
B
and

~a

=
1
2

CB

.
4.
Given two vectors
~a
=
h
1
,
2
,
1
i
and
~
b
=
h
1
,
1
,
2
i
ﬁnd
a) the angle between
~a
and
~
b
;
Answer.
For the angle
α
between them
cos
α
=
~a
·
~
b

~a
 
~
b

=
3
6
=
1
2
,
We have
~a
·
~
b
=

1+2+2 = 3
,

~a

=
√
1 + 4 + 1 =
√
6 =
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 '07
 KAMIENNY
 Math, Calculus

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