3(x  (1))  2(y  2) + (z  (3)) =
0
______________________________________________________________________
12. (5 pts.) Which point on the line defined the vector equation
<x,y,z> = <1,1,1> + t<2,1,1> is nearest the point, (0,1,0)?
Build the vector with initial point (0,1,0) to an arbitrary point on
the line with parameter
t
,
The norm of this vector is the distance from the point (0,1,0) to the
point on the line with parameter value
t
.
The norm will be smallest when
the vector is perpendicular to the vector <2,1,1>.
Consequently,
provides us with the value of
t
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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______________________________________________________________________
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:
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 Spring '06
 GRANTCHAROV
 Multivariable Calculus, pts, Euclidean geometry

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