Solutions to Exercises
Problem Set 1.1, page 6
1
Line through (1
,
1
,
1); plane; same plane!
3
v
= (2
,
2) and
w
= (1
,

1).
4
3
v
+
w
= (7
,
5) and
v

3
w
= (

1
,

5) and
c
v
+
d
w
= (2
c
+
d, c
+ 2
d
).
5
u
+
v
= (

2
,
3
,
1) and
u
+
v
+
w
= (0
,
0
,
0) and 2
u
+2
v
+
w
= (add first answers) = (

2
,
3
,
1).
6
The components of every
c
v
+
d
w
add to zero. Choose
c
= 4 and
d
= 10 to get (4
,
2
,

6).
8
The other diagonal is
v

w
(or else
w

v
). Adding diagonals gives 2
v
(or 2
w
).
9
The fourth corner can be (4
,
4) or (4
,
0) or (

2
,
2).
10
i
+
j
is the diagonal of the base.
11
Five more corners (0
,
0
,
1)
,
(1
,
1
,
0)
,
(1
,
0
,
1)
,
(0
,
1
,
1)
,
(1
,
1
,
1). The center point is (
1
2
,
1
2
,
1
2
). The
centers of the six faces are (
1
2
,
1
2
,
0)
,
(
1
2
,
1
2
,
1) and (0
,
1
2
,
1
2
)
,
(1
,
1
2
,
1
2
) and (
1
2
,
0
,
1
2
)
,
(
1
2
,
1
,
1
2
).
12
A fourdimensional cube has 2
4
= 16 corners and 2
·
4 = 8 threedimensional sides and 24
twodimensional faces and 32 onedimensional edges. See Worked Example
2.4 A
.
13
sum = zero vector; sum =

4:00 vector; 1:00 is 60
◦
from horizontal = (cos
π
3
,
sin
π
3
) = (
1
2
,
√
3
2
).
14
Sum = 12
j
since
j
= (0
,
1) is added to every vector.
15
The point
3
4
v
+
1
4
w
is threefourths of the way to
v
starting from
w
. The vector
1
4
v
+
1
4
w
is
halfway to
u
=
1
2
v
+
1
2
w
, and the vector
v
+
w
is 2
u
(the far corner of the parallelogram).
16
All combinations with
c
+
d
= 1 are on the line through
v
and
w
. The point
V
=

v
+ 2
w
is
on that line beyond
w
.
17
The vectors
c
v
+
c
w
fill out the line passing through (0
,
0) and
u
=
1
2
v
+
1
2
w
. It continues
beyond
v
+
w
and (0
,
0). With
c
≥
0, half this line is removed and the “ray” starts at (0
,
0).
18
The combinations with 0
≤
c
≤
1 and 0
≤
d
≤
1 fill the parallelogram with sides
v
and
w
.
19
With
c
≥
0 and
d
≥
0 we get the “cone” or “wedge” between
v
and
w
.
20
(a)
1
3
u
+
1
3
v
+
1
3
w
is the center of the triangle between
u
,
v
and
w
;
1
2
u
+
1
2
w
is the center
of the edge between
u
and
w
(b) To fill in the triangle keep
c
≥
0,
d
≥
0,
e
≥
0, and
c
+
d
+
e
= 1.
3