18.02, Spring 2008
Practice Test 1
The first test will take place on
Thursday, 21 February
and will cover the material of lectures
one to six. The problems below should give you a sense of the level of difficulty and length of the
test—you do not need to turn them in. The solutions will be made available to you on Monday,
18 February.
There will be a review session for the test on
Wednesday, 20 February
at 7:30 p.m. in 4163.
1.
(a) Find the area of the triangle with vertices
P
1
= (2
,
1
,
0),
P
2
= (1
,
1
,
1), and
P
3
=
(0
,

1
,
3).
(b) Find the equation of the plane containing the points
P
1
,
P
2
, and
P
3
.
(c) Find the point where the line through
Q
1
= (

2
,
1
,
3) and
Q
2
= (0
,

3
,
1) intersects the
plane found in (b).
2. Find all the values of
c
for which the planes
2
x

y
+
z
= 0
4
x

2
y
+
z
= 0
cx
+
y
= 1
meet in a unique point.
3.
(a) Find the point of intersection of the three planes
2
x
+
y

2
z
= 1
x

y

z
= 3
x
+ 2
y
+
z
= 2
by writing the equations above as a linear system
Ax
=
b
, finding the inverse of
A
, and
computing
A

1
b
.
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 Spring '08
 LUCAS
 Multivariable Calculus, Vectors, Acceleration, Velocity, acceleration vector

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