\mcv4U\vectors\collinear and coplanar.doc
MCV 4U Collinear and Coplanar Vectors
1. Two vectors are collinear if one is a scalar multiple of the other.
Ex. If
a
k
b
= , where ‘
k
‘ is a real number,
then
b
a
&
are collinear
2. i) Any two (2) non-collinear vectors are coplanar.
ii) If three non-collinear vectors are coplanar, any one of them can be expressed as a
linear combination of the other two.
i) ii)
3. Non-coplanar vectors
4. Linearly Dependent vectors.
i) collinear in 2-space,
R
R
2
2
b
k
a
= , where
ℜ∈
k
i
i
i
i
)
)
coplanar in 3-space,
R
R
3
3
b
t
a
s
c
+= , where
ℜ
∈
t
s
,
5. Linearly Independent vectors
No simple equation relating the vectors either linearly or in the same plane
6. Example
Given:
]
[
4
,
1
,
3
−=
a
,
]
[
8,4,6 −−=
b
and
]
[
4
,
3
,
7
−=
c
.
Find: Determine whether
ba
,
and
c
form a basis of a plane (ie. whether they are
coplanar and can be written as a linear combination).
*We must first check whether they’re non-collinear (ie. none are scalar multiples of others)
cmbka
,≠
∴none are collinear
Sol’n: Let
s
and
t
be scalars such that
b
t
a
s
c
+=
.
][ ][ ][
][ ][ ][
][ ][
tststs
tttsss
ts
btasc
84,4,634,3,7
8,4,64,1,34,3,7
8,4,64,1,34,3,7
−−−+=−
−−+−=−
−−+−=−
+=
a
b
k
= 2
a
b
a
b
c
All on the
same
surface
a
b
c
3
1
3
5
,
==
t
s
∵
satisfy all three
equations, the linear combination
exists and vectors
cba
,,
are
coplanar (ie. form a basis of a plane)
\mcv4U\vectors\collinear and coplanar.doc
7. Test for 3 points being Collinear (p. 89 #13b (i))
P
(2, 1, -3),
Q
(-4, 5, -1) and
R
(5, -1, -4)
Does
PRsPQ
=
, where
s
is a scalar?
]
[
2,4,6−=
PQ
and
]
[
1,2,3 −−=
PR
]
[
PRPQ
PQPR
PR
PR
2
2
1
2,4,6
2
1
2
2
,
2
4
,
2
6
−=
−=
−−=
−−−
−=
∴
P
,
Q
, and
R
are collinear.
8. Test for 4 points being Coplanar (p. 90 #18a,b (i))
a) Form 3 vectors and test whether a linear combination exists.
b) (i)
A
(3, 1, 0),
B
(2, -3, 1),
C
(-1, 0, 4),
D
(5, -6, -2)
]
[
][
][
2,7,2
4,1,4
1,4,1
−−=
−−=
−−=
AD
AC
AB
]
[
[
]
]
[
]
[ ][
tststs
t
s
24,7,241,4,1
2
,
7
,
2
4
,
1
,
4
1
,
4
,
1
−−−+−=−−
−
−
+
−
−
=
−
−
Solve for
s
and
t
and determine
∴
ADACAB
,,
are coplanar, so
A
,
B
,
C
and
D
must also be coplanar.
Practice:
Let
s
and
t
be scalars such that
AD
t
AC
s
AB
+=
