Theorem
Statement of the theorem
If and be any two non-zero and non- collinear plane vectors then any vector in their plane can be uniquely expressed as the sum of the vectors parallel to the vectors and .
Proof: Let , and be the vectors acting at the point O as shown in the figure. Draw PH and PG parallel to OB and OA respectively. By doing so, we obtain a parallelogram OHPG. Now by the parallelogram law of addition of vectors, or ......................(i). But and are respectively parallel to the vectors and . So we can write and . Putting these values in the expression (i), we get, which is the required result to prove the statement of the above theorem. For uniqueness, if possible, let ..............(ii). From (i) and (ii), we get, or ()() =0 which is possible only when or and . Hence the expression (i) is unique.
Coplanar and non-coplanar vectors
Any finite number of vectors are said to be coplanar if a plane can be drawn parallel to all of them. If this is not the case, then they are called non-coplanar vectors. As it is always possible to draw a parallel plane to any number of two-dimensional vectors, so all the two dimensional vectors are always coplanar.
Three-dimensional vectors or space vectors: All vectors other than the two dimensional vectors are called the space vectors. The space vectors do not lie on the same plane.
Theorem on Three dimensional vectors: If be any three non-zero and non-collinear vectors and x,y,z be any three scalars then implies x = y = z = 0.
Coplanar and non-coplanar vectors
Any finite number of vectors are said to be coplanar if a plane can be drawn parallel to all of them. If this is not the case, then they are called non-coplanar vectors. As it is always possible to draw a parallel plane to any number of two-dimensional vectors, so all the two dimensional vectors are always coplanar.
Three-dimensional vectors or space vectors: All vectors other than the two dimensional vectors are called the space vectors. The space vectors do not lie on the same plane.
Theorem on Three dimensional vectors: If be any three non-zero and non-collinear vectors and x,y,z be any three scalars then implies x = y = z = 0.