Linearly Independent and Dependent System of Vectors
( i) The system of vectors a, b, c,… is said to be linearly dependent, if there exists a scalars x, y, z, … not all zero, such that xa + yb + zc + … = 0.
( ii) The system of vectors a, b, c, … is said to be linearly independent, if xa + yb + zc + td = 0 rArr; x + y + z + t … = 0.
Important Points to be Remembered( i) Two non-collinear vectors a and b are linearly independent.
( ii) Three non-coplanar vectors a, b and c are linearly independent.
( iii) More than three vectors are always linearly dependent.
Resolution of Components of a Vector in a Plane
Let a and b be any two non-collinear vectors, then any vector r coplanar with a and b, can be uniquely expressed as r = x a + y b, where x, y are scalars and x a, y b are called components of vectors in the directions of a and b, respectively.
∴ Position vector of P( x, y) = x i + y j. OP 2 = OA 2 + AP 2 = | x | 2 + | y | 2 = x 2 + y 2
OP = √x 2 + y 2. This is the magnitude of OP.
where, x i and y j are also called resolved parts of OP in the directions of i and j, respectively.