TEST1/MAC2313
Page 5 of 5
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17. (5 pts.) The point (5,5 3,10) is in rectangular coordinates.
Convert
this to spherical coordinates (
ρ
,
θ
,
φ
).
(
ρ
,
θ
,
φ
) = ( 10 (2)
1/2
, 5
π
/3,
π
/4)
.
ρ
2
= 5
2
+(5 3)
2
+ 10
2
= 200
φ
= cos
1
(1/(2)
1/2
) =
π
/4
θ
has its terminal side in the 4th quadrant, and its reference angle is
θ
r
= tan
1
( 3) =
π
/3.
Consequently
θ
= 2
π

θ
r
= 5
π
/3.
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18. (5 pts.) Do the lines defined by the equations
<x,y,z> = <3,1,2> + t<2,1,2>
and
<x,y,z> = <10,8,5> + t<1,3,1>
intersect?
If they do intersect, what is the point of intersection??
If the lines are to intersect, there must be numbers t
1
and t
2
so
3 + 2t
1
= 10 + t
2
, 1  t
1
= 8 + 3t
2
, and 2  2t
1
= 5  t
2
.
[Here t
1
is the
parameter for the putative point in terms of the first equation and t
2
is
the parameter value for the second equation.]
Solving this system yields
t
1
= 2 and t
2
= 3.
Using either t
1
or t
2
in the appropriate vector
equation yields the point of intersection, (7,1,2).
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19. (5 pts.) What is the area of the triangle in three space with vertices
at P = (1, 0, 0), Q = (0, 2, 0), and R = (0 , 0, 3).
Let
v
be the vector with initial point P and terminal point Q, and let
w
be the vector with initial point P and terminal point R. Then the area A
of the triangle is given by
A =
v
×
w
/2 =
<1,2,0> × <1,0,3> /2 =
<6,3,2> /2 = 7/2.
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20. (5 pts.) Do the three 2space sketches of the traces in each of the
coordinate planes of the surface defined by
y
= 1 
x
2

z
2
. Work below and
label carefully.
Then attempt to do a 3  space sketch in the plane of
the surface on the back of page 4.