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?
This
preview
has intentionally blurred sections.
Sign up to view the full version.
This is the end of the preview.
Sign up
to
access the rest of the document.
