Spring 2012 Math 263 Test 1 Name: _____________________
Show all necessary work neatly, clearly, systematically, and understandably. Any incorrect statement and/or understatement may be penalized. There are 106 points available.
1.
(19: 2,3,3,3,2,3,3)
Let A

2 ,

3 , 1

, B


1 , 2 , 0

and C

1 ,

2 , 3

be coordinates in space. Find:
Let v

AB and w

AC a.
v and w d.
Proj w v b.
the equation of the line AB e.
the area of

ABC c.
v
 w f.
the distance from C to line AB g.
the equation of the plane containing

ABC

2.
(10:3,4,3)
Let P
1
: 4 x

3 y

2 z
 a.
Angle between the planes
5 and P
2
: 2 x

3 y
 z

1 . Find b.
The equation of the line formed by the intersection of the planes. c.
The angle formed by the line and the xy-plane.
3.
(30: 2,2,2,2,2,2,2,3,3,3,3,4)
Let r ( t )
 a sin t , a cos t , bt . Find: a.
r ' ( t ) c.
s
0 b.
r ' ( t ) d.
Reparametrize r ( t ) by s
0

e.
T ( t ) f.
T ' ( t ) g.
   h.
N ( t ) i.
B ( t ) j.
The equation of the line tangent to r ( t ) at t


2 k.
The equation of osculating plane at t


2
. l.
Suppose a ( t )
 a
T
T
 a
N
N , where a
T and a
N
are tangential and normal t components of the acceleration, respectively. Find a
T
and a
N
at general
. Note: a ( t )
 d
2 s
T dt
2
 ds dt
2

N

4.
(17: 3,3,5,6) a.
Consider the following surfaces:
Convert the equation of equation.
S
1
S
1
:
 
to rectangular

4
and S
2
: r
2  z . d.
Find the equation for the intersection of the surfaces. Note: Use parametric equations. b.
Convert the equation of S
2
to rectangular equation. c.
Sketch the surfaces and their intersection.
5.
(6)
Graph 4 x
2  y
2 
9 z
2 
36 . Denote the intercepts on the graph

6.
(10:4,6)
Consider lines r
1
( t )

3

2 t ,

1

4 t , 2
 t and r
2
( s )

3

2 s , 2
 s ,

2

2 s a.
Determine whether the lines are intersecting or skew. b.
If those lines are intersecting, find the intersecting points. Otherwise, find the distance between those lines.
7.
(14:4,4,6)
Consider the curve r ( t )
 t , t
2
,

3 t and the plane P : x
 y
 z

3 . a.
At what t does the curve hit the plane? There are 2 solutions b.
Find the coordinates of the intersections c.
At what angles does the curve hit the plane?