Homework 1 Solutions
Josh Hernandez
October 25, 2009
1.1  Introduction
2. Find the equations of the lines through the following pairs of points in space.
b. (3
,
2
,
4) and (5
,
7
,
1)
Solution:
x
= (3
,
2
,
4) +
r
[(5
,
7
,
1)

(3
,
2
,
4)] = (3
,
2
,
4) +
r
(8
,
9
,
3)
.
d. (2
,
1
,
5) and (3
,
9
,
7)
Solution:
x
= (2
,
1
,
5) +
r
[(3
,
9
,
7)

(2
,
1
,
5)] = (2
,
1
,
5) +
r
(5
,
10
,
2)
.
3. Find the equations of the planes containing the following points in space.
b. (3
,
6
,
7), (2
,
0
,
4), and (5
,
9
,
2).
Solution:
x
= (3
,
6
,
7) +
r
[(2
,
0
,
4)

(3
,
6
,
7)] +
s
[(5
,
9
,
2)

(3
,
6
,
7)]
= (3
,
6
,
7) +
r
(5
,
6
,
11) +
s
(2
,
3
,
9)
d. (1
,
1
,
1), (5
,
5
,
5), and (6
,
4
,
2).
Solution:
x
= (1
,
1
,
1) +
r
[(5
,
5
,
5)

(1
,
1
,
1)] +
s
[(6
,
4
,
2)

(1
,
1
,
1)]
= (1
,
1
,
1) +
r
(4
,
4
,
4) +
s
(7
,
3
,
1)
7. Prove that the diagonals of a parallelogram bisect each other.
Solution:
A parallelogram
ABCD
in
R
2
may be translated so that vertex
A
lies on the origin.
After translation, denote the coordinatevectors of the adjacent vertices
B
and
D
by
v
and
w
.
The four vertices therefore have coordinatevectors 0
,v,v
+
w
, and
w
.
By the midpoint rule, segment
BD
has midpoint at
1
2
(
v
+
w
), and
AC
has midpoint
1
2
(0+(
v
+
w
)).
These midpoints are identical, so
BD
and
AC
bisect one another.
1.2  Vector Spaces
2. Write the zero vector of
M
3
×
4
(
F
).
1