MATH 214 A1  Assignment 4
Due Friday, June 3, 5:00pm
(1) Use inequalities to describe the solid cube in the ﬁrst octant
bounded by the coordinate planes and the planes
x
= 2
,y
= 2
and
z
= 2. (2 marks)
{
(
x,y,z
)

0
≤
x
≤
2
,
0
≤
y
≤
2
,
0
≤
z
≤
2
}
(2) Write the following vectors as a product of their length and
direction: (2 marks each)
(a)
h
9
,

2
,
6
i
11
· h
9
11
,

2
11
,
6
11
i
(b)
h
1
√
3
,
1
√
3
,
1
√
3
i
1
· h
1
√
3
,
1
√
3
,
1
√
3
i
(c)
h
1
,

1
,
1
i
√
3
h
1
√
3
,

1
√
3
,
1
√
3
i
(3) Write down the equation for the four planes that bound the
tetrahedron with corners (1
,
0
,
0)
,
(0
,
1
,
0)
,
(0
,
0
,
1) and (0
,
0
,
0).
(4 marks)
x
= 0,
y
= 0,
z
= 0 and
x
+
y
+
z
= 1
(4) Show that for any vector
v
, the projection of
v
onto
v
is
v
. (3
marks)
v
·
v

v

v

·
v
=

v

2

v

v

·
v
= 1
·
v
=
v
(5) Consider a sailboat being moved by the wind. If the wind exerts
a force of 100 N due north and the boat travels at an angle 60
0
north of east, how much work is done by the wind in moving