42
CHAPTER 5.
LINES AND PLANES
using the expansion of the determinant along the first row.
T5.
Show that
u
and
v
are parallel if and only if
u
×
v
=
0
.
Solution.
In Section 4.2 (p.
238), the following definition is given:
two nonzero vectors
u
and
v
are
parallel
if

u
•
v

=
k
u
k k
v
k
(that is,
cos
θ
=
±
1, or, equivalently sin
θ
= 0,
θ
denoting the angle of
u
and
v
).
Notice that
k
u
k 6
= 0
6
=
k
v
k
for nonzero vectors and
u
×
v
=
0
⇔
k
u
×
v
k
= 0.
Using the length formula
k
u
×
v
k
=
k
u
k k
v
k
sin
θ
we obtain
sin
θ
= 0 if and only if
u
×
v
=
0
, the required result.
Page 271.
T5.
Show that an equation of the plane through the noncollinear points
P
1
(
x
1
, y
1
, z
1
)
, P
1
(
x
2
, y
2
, z
2
) and
P
1
(
x
3
, y
3
, z
3
) is
x
y
z
1
x
1
y
1
z
1
1
x
2
y
2
z
2
1
x
3
y
3
z
3
1
= 0
.
Solution.
Any three noncollinear points
P
1
(
x
1
, y
1
, z
1
)
, P
1
(
x
2
, y
2
, z
2
) and
P
1
(
x
3
, y
3
, z
3
) determine a plane whose equation has the form
ax
+
by
+
cz
+
d
= 0
,
where
a
,
b
,
c
and
d
are real numbers, and
a
,
b
,
c
are not all zero.
Since
P
1
(
x
1
, y
1
, z
1
),
P
1
(
x
2
, y
2
, z
2
) and
P
1
(
x
3
, y
3
, z
3
) lie on the plane, their coordi
nates satisfy the previous Equation:
ax
1
+
by
1
+
cz
1
+
d
= 0
ax
2
+
by
2
+
cz
2
+
d
= 0
ax
3
+
by
3
+
cz
3
+
d
= 0
.