needed for the closest point.
The point in
question may now be obtained using the vector equation for the line. It is
(8/6,5/6,5/6).
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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  3y
= 5
and
2y  4z = 7.
θ
= cos
1
((
v w
)/(
v
w
)) = cos
1
(6/(200)
1/2
) = cos
1
(3/(5(2)
1/2
))
where
v
= <1,3,0> and
w
= <0,2,4>.
Observe acutely the funny absolute
value thingies.
______________________________________________________________________
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.
A normal vector
n
for the plane sought is
since it must be perpendicular to a direction vector for the line and a
normal vector of the given plane.
[One may also obtain a suitable vector
by solving an appropriate system of equations, a little two by three linear
homogeneous thingy.]
An equation for the plane is now cheap thrills:
2(
y
 2)  4(
z
 3) = 0, or equivalently,
y
+ 2
z
 8 = 0.
______________________________________________________________________
15. (5 pts.) Obtain an equation for the plane tangent to the sphere defined
by
(
x
1)
2
(
y
2)
2
(
z
3)
2
9
at the point (2, 4, 5), which is actually on the sphere.
Here all we need is a normal vector for the tangent plane.
This can
be obtained easily using the point of tangency and the center of the sphere
at hand.
An equation of the tangent plane:
(
x
 2) + 2(
y
 (4))  2(
z
 5) = 0.
You may obtain an equivalent standard form varmint if you wish.
______________________________________________________________________
16. (5 pts.)
The equation
r
4cos(
θ
)
10sin(
θ
)
is that of a cylinder in cylindrical coordinates.
Obtain an equivalent
equation in terms of rectangular coordinates (x,y,z).
Provide a vector
equation for the straight line that is the axis of symmetry.
Multiply the given equation by
r
.
Then doing the usual conversion and
completing the square a couple of times yields
(
x
2)
2
(
y
5)
2
29.
A vector equation for the line of symmetry in 3space is
<
x
,
y
,
z
>
<2,
5,
t
>,
for t
∈
.
TEST1/MAC2313
Page 5 of 5
______________________________________________________________________
17. (5 pts.) The point (3,4,5) is in rectangular coordinates.
Convert
this to spherical coordinates (
ρ
,
θ
,
φ
).
[Inverse trig fun?]
______________________________________________________________________
18. (5 pts.) Do the lines defined by the equations
<x,y,z> = <0,1,2> + t<4,2,2> and <x,y,z> = <1,1,1> + t<1,1,4>
intersect?
Justify your answer, for
yes
or
no
does not suffice.
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 Spring '06
 GRANTCHAROV
 Multivariable Calculus, pts, Euclidean geometry

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