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 .