Math 241 – Exam 1 – Version 1
September 26, 2007
50 points possible
1.
Consider the points
A
(5
,

4
,
7) and
B
(3
,
0
,
1).
(a)
(3pts) Find a vector equation or a set of parametric equations that defines the line
containing points
A
and
B
.
(b)
(2pts) Determine if the line from part (a) intersects the plane 3
x
+ 2
y

7
z
=

86.
2.
(a)
(2pts)If
a
= (1
,

1
,
7
,
3
,
2) and
b
= (2
,
5
,
0
,
9
,

1), calculate proj
b
a
.
(b)
(3pts) Under what circumstances is proj
a
b
= proj
b
a
? Briefly explain your answer.
3.
(a)
(3pts) State the dot product definition for the length of a vector in
R
n
.
(b)
(3pts) Clearly state the CauchySchwarz Inequality.
(c)
(4pts) Let
a
and
b
be vectors in
R
n
.
Use CauchySchwarz to prove the Triangle
Inequality

a
+
b
 ≤ 
a

+

b

.
(Hint: Start with

a
+
b

2
.)
4.
Let
u
= (1
,
2
,

1),
v
= (3
,
0
,

1) and
w
= (

5
,
4
, b
).
(a)
(3pts) Find the volume of the parallelopiped determined by
u
,
v
and
w
.
(b)
(2pts) For what values of
b
are the vectors
u
,
v
and
w
coplanar?
5.
Consider the lines
l
1
:
x
=
t

3
, y
= 1

2
t, z
= 2
t
+ 5 and
l
2
:
x
= 4

2
t, y
= 4
t
+ 3
, z
=
6

4
t
.
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 Spring '08
 Kim
 Math, Linear Algebra, Equations, Dot Product, Parametric Equations, Euclidean vector, Parametric equation, Inner product space

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