MATH 51 MIDTERM 1
January 29, 2004
1.
Find all solutions of the following system:
x
1
-
x
2
+
x
3
+
2
x
4
=
3
x
2
+
x
3
+
x
4
=
3
x
1
+
x
2
+
3
x
3
+
4
x
4
=
9
2.
Let
L
be the intersection of the two planes
x
+
y
+
z
= 4
and
2
x
+ 3
y
+
z
= 9
.
Find a parametric equation for
L
.
3(a)
Suppose
u
,
v
, and
w
are points in
R
n
such that
k
u
k
=
k
v
k
=
k
w
k
= 1 and
such that
w
=
-
u
. Suppose also that
v
is not equal to
u
or to
w
. Prove that the
triangle Δ
uvw
has a right angle at
v
.
3(b)
Suppose
x
,
y
, and
z
are vectors in
R
n
whose norms are 1, 2, and 3, respec-
tively. Suppose each vector is orthogonal (i.e., perpendicular) to each of the other
two. Find a scalar
c
such that the vector
x
+
c
y
-
z
is orthogonal to the vector
x
+
y
+
z
.
4.
Consider the points
A
= (1
,
1
,
1
,
1),
B
= (1
,
2
,
0
,
-
1) and
C
= (1
,
0
,
-
1
,
1) in
R
4
.
4(a)
Find the cosine of the angle at
B
of the triangle
ABC
.
4(b)
Find a parametric equation for the plane through the points
A
,
B
, and
C
from part (a).
5.
Are the following three vectors in
R
3
linearly independent or linearly dependent?
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