Collinear and
Coplanar Vectors
Collinearity
If
a vector
is multiplied by a scalar k then its
magnitude changes k times. The direction of such a vector remains same if k
> 0 however it reverses if k < 0 and becomes indeterminate if k = 0. So
by multiplying with k we obtain a vector which has same line of support as
vector
. These two vectors are called
collinear vectors. Note here that null vector is collinear to all the vectors.
Collinearity and linear
dependence
If
two vectors
and
are
collinear then they are linearly dependent and vice versa. On the other hand if
two vectors
are non–collinear then they are linearly
independent and vice versa. In this case
Û x = 0 and y = 0
Methods to prove
collinearity:
- Two
vectors
and
are
collinear if there exists k ∈
R such that
. - If
are collinear. - Three
points A(
), B(
),
C(
)are collinear if there exists k∈R such that
that
is
. - If
then A, B, C are collinear. - A(
), B(
),
C(
) are collinear if there exists
scalars l, m, n, (not all zero) such that
where
l + m + n = 0. - All
the above methods are equivalent and any of them can be utilized to prove the
collinearity. (of 2 vectors or 3 points).
Coplanarity
- Two
vectors (free) are always coplanar.
- Two
non-collinear vectors
always determine a unique plane. Hence any vector in that plane can be uniquely
represented as a linear combination of these two vectors. .
(obviously vectors
are coplanar).
Methods to prove
coplanarity:
- Three
vectors
are coplanar if there exists l, m∈R such that
i.e.,
one can be uniquely expressed as a linear combination of the other two. - If
are coplanar (necessary and
sufficient condition). - Four
points A
, B
, C
and D
lie
in the same plane if there exist l, m∈R
such that
i.e.
. - If
= 0 then points A, B, C, D are
coplanar. - A,
B, C, D are coplanar if there exists scalars k, l, m, n (not all zero), such
that
where k + l + m + n = 0.
Example.1
If the vectors
,
and
are
coplanar (where x, y, z > 1) then prove that
=1.
Solution
Since vectors are coplanar we can find l and m such that
= l
(
) + m(
)
\
1 –x = 1 – l – m, y =
,
z = ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_8_files/image037.gif)
1 – y =
, 1 – z = ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_8_files/image039.gif)
\
= ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_8_files/image041.gif)
=
= 1.
Example.2
Vectors
are such that
every pair is non collinear and the vector
is
collinear with
and the vector
is collinear with
. Prove that
.
Solution
Given that
....(1)
and
....(2)
\
( putting that value of
from (1) and (2) )
&⇒
...(3)
\
are non collinear so they are
linearly independent hence (3) will hold only if 1 + l = 0 and 1 + m = 0
&⇒ l = m = –1
Using
the value of l in (1) we get
.