Math 20C – Prof. Rabin – First Midterm Exam, Version A – October 22, 2007
Write your name and your section number or time at the top of this exam. You may not use
notes, a calculator, or any other assistance.
If any question is not clear, ask the instructor for
clarification. Show all work necessary to justify your answers. Please box your answers and cross
out any “false starts”. Each problem is worth 25 points. Good luck!
(1) (a) Find the equation of the plane that contains the points
A
(6
,
2
,
1),
B
(2
,
4
,
0), and
C
(2
,
2
,

1).
Answer:
Two (displacement) vectors in the plane are
~
AB
=
h
4
,
2
,

1
i
and
~
AC
=
h
4
,
0
,

2
i
.
The normal vector to the plane can now be calculated as
~n
=
~
AB
×
~
AC
=
i
j
k

4
2

1

4
0

2
=
h
4
,

4
,
8
i
.
For simplicity, use 
1
4
~n
=
h
1
,
1
,

2
i
instead of
~n
.
Then using the point
B
= (2
,
4
,
0) and

~n
4
, we
have the equation
1
·
(
x

2) + 1
·
(
y

4) + 2
·
(
z

0) = 0
,
so the equation of the plane is
x
+
y

2
z
= 6
.
(b) Find the distance from the origin to this plane.