Linear
combination of vectors
The
linear combination of a finite set of vectors …is defined as a vector such that = + + …+
,
where
k1, k2, … kn are any scalars (real numbers).
Fundamental theorem in
two-dimensions:
Let and be two non-zero non-collinear vectors.
Then any vector in the plane of and can
be expressed uniquely as a linear combination of and
i.e. there exist unique l, m∈R
such that l+m= .
This
also means that if l1+ m1= l2+ m2then l1 = l2
and m1 = m2.
Fundamental theorem in
three-dimensions
Let , and be
three non-zero non-coplanar vectors in space. Then any vector in space can be expressed uniquely as
a linear combination of , and i.e
there exist unique l, m, n ∈R
such that l+ m+
n=
This also means that if l1+ m1+ n1=
l2 + m2+ n2then
l1 = l2, m1 = m2 and
n1 = n2
Linearly dependent and
independent vectors:
A system of vectors is
said to be linearly dependent if there exists a system of scalars k1,
k2 …, kn (not all zero) such that k1+ k2+ … +kn=
They are said to be linearly independent if every
relation of the type
k1+ k2+ … +kn= implies
that k1 = k2 =….=kn = 0
Note:
Two
collinear vectors are always linearly dependent.
Two
non-collinear non-zero vectors are always linearly independent
Three
coplanar vectors are always linearly dependent.
Three
non-coplanar non-zero vectors are always linearly independent.
More
than 3 vectors are always linearly dependent.
Three
points with position vectors are collinear if l1with l1 + l2 + l3 = 0.
Four points with position vectors ,are
coplanar if l1with l1 + l2 + l3 + l4= 0.
Example.1
Show
that points with p.v. are
collinear. It is given that vectors are non-coplanar.
Solution
The three points are collinear, if we can
find l1, l2 and l3, such that
l1
with l1 + l2 +l3 = 0.
Equating the coefficients separately to zero, we get
l1 – 2l2 + 4l3 = 0,
-2l1 + 3l2 - 7l3 = 0 and 3l1 - l2 + 7l3 = 0
We find that l1 = -2, l2 = 1, l3 = 1 so that l1 + l2 + l3 = 0
Hence the given vectors are collinear.