Consequently, a simple pointnormal equation for the plane is given by
2(x4)  3(y+5)  3(z6) = 0.
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TEST1/MAC2313
Page 4 of 5
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13. (5 pts.) Find the exact value of the acute angle
θ
of intersection of
the two planes defined by the two equations
x  2y
= 55
and
3y  4z = 75.
θ
= cos
1
((
v w
)/(
v
w
)) = cos
1
(6/(125)
1/2
)
where
v
= <1,2,0> and
w
= <0,3,4>.
Observe acutely the funny absolute
value thingies.
Why do we need them generally to get acute angles??
_________________________________________________________________
14. (5 pts.) Write an equation for the plane which contains the line
defined by <x,y,z> = <1,2,3> + t<3, 2, 1> and is perpendicular to the
plane defined by x  2y + z = 0.
Since a normal vector
v
for the plane must be perpendicular to both
the direction of the line, due to the line lying in the desired plane, and
any nonzero normal vector for the given plane defined by the equation
x  2y + z = 0, we may build
v
by means of a cross product.
Set
v
= <3,2,1> × <1,2,1> = <0,2,4>.
A pointnormal equation for the plane
we want is now cheap thrills:
2(y  2)  4(z  3) = 0 ... ugh.
_________________________________________________________________
15. (5 pts.) What is the radius of the sphere centered at
(1, 0, 0) and tangent to the plane defined by x + 2y + z = 10?
The radius,r , of the sphere will simply be the distance from the
point (1,0,0) to the plane defined by the equation x  2y + z  10 = 0.
Using the appropriate formula, we have
r =
(1)(1)+(2)(0)+(1)(0)10 /(1 + 4 + 1)
1/2
=
9/(6)
1/2
.
_________________________________________________________________
16. (5 pts.) The equation
z
= 3
r
2
cos
2
(
θ
)
is in cylindrical coordinates.
Obtain an equivalent equation in terms of
rectangular coordinates (x,y,z).
Wake me up ... z = 3x
2
.
TEST1/MAC2313
Page 5 of 5
_________________________________________________________________
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.
_________________________________________________________________
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??
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 Spring '06
 GRANTCHAROV
 Multivariable Calculus, pts, Euclidean geometry

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