MATH 51 MIDTERM 1 SOLUTIONS
1. Find all solutions of the following system:
x
1
+
x
2
+
x
4
=
7
x
1
+
x
2
+
x
3
+
x
4
=
10
x
1
+
x
3
+
x
4
=
9
Solution.
The augmented matrix for this system is
1
1
0
1
1
1
1
1
1
0
1
1
7
10
9
Its reduced row echelon form is
1
0
0
1
0
1
0
0
0
0
1
0
6
1
3
so the reduced form of the system is
x
1
+
x
4
=
6
x
2
=
1
x
3
=
3
Solving for the pivot variables
x
1
, x
2
, x
3
we find
x
=
x
1
x
2
x
3
x
4
=
6

x
4
1
3
x
4
=
6
1
3
0
+
x
4

1
0
0
1
2. Let
L
be the intersection of the two planes
x
+ 2
y
+ 3
z
= 10
and
4
x
+ 5
y
+ 6
z
= 28
.
Find a parametric equation for
L
.
Solution.
The augmented matrix for this system of equations is
1
2
3
4
5
6
10
28
Its reduced row echelon form is
1
0

1
0
1
2
2
4
1
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so the system reduces to
x

z
=
2
y
+
2
z
=
4
Solving for the pivot variables
x
and
y
gives
x
y
z
=
2 +
z
4

2
z
z
=
2
4
0
+
z
1

2
1
3. (a) Suppose
u
and
v
are vectors in
R
n
such that
u
+
v
and
u

v
are orthogonal (i.e.,
perpendicular) to each other. Show that
k
u
k
=
k
v
k
.
Solution.
Two vectors are orthogonal if and only if their dot product is zero. So
(
u
+
v
)
·
(
u

v
) = 0
The left hand side expands to
u
·
u

u
·
v
+
v
·
u

v
·
v
=
u
·
u

v
·
v
=
k
u
k
2
 k
v
k
2
Thus
k
u
k
2
=
k
v
k
2
, so
k
u
k
=
k
v
k
.
(b) Suppose
u
,
v
, and
w
are unit vectors in
R
n
. (Recall that a unit vector is a vector
whose length is 1.) Suppose each vector is orthogonal (i.e., perpendicular) to each
of the other two. Show that the two vectors
u

3
v
+ 2
w
and
u
+
v
+
w
are orthogonal to each other.
Solution.
Since
u
,
v
and
w
are unit vectors, their lengths (and hence their lengths
squared) are all equal to 1. So
u
·
u
=
v
·
v
=
w
·
w
= 1. Since each vector is
perpendicular to the others,
u
·
v
=
v
·
u
= 0,
u
·
w
=
w
·
u
= 0 and
v
·
w
=
w
·
v
= 0.
So the dot product of the two vectors given is
(
u

3
v
+ 2
w
)
·
(
u
+
v
+
w
) =
u
·
u
+
u
·
v
+
u
·
w

3
v
·
u

3
v
·
v

3
v
·
w
+ 2
w
·
u
+ 2
w
·
v
+ 2
w
·
w
= 1

3 + 2 = 0
so they are orthogonal.
4. Consider the points
A
= (1
,
1
,
1),
B
= (1
,
3
,
1) and
C
= (1
,
1
,
4) in
R
3
.
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 '07
 Staff
 Math, Linear Algebra, Algebra, Differential Calculus, ax, Row echelon form

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