1.Show that the vectors A, B, C  with position vectors  and   are collinear.

2.Prove that vector addition is commutative that is, if and are any two vectors then

3.Prove that vector addition is associative.

4.Prove that the medians of a triangle are concurrent.

Let ABC be a triangle and let D, E, F be the midpoints of its sides BC, CA and AB respectively. We have to prove that the medians AD, BE, CF are concurrent.

Let O be the origin and be the position vectors of A, B, C respectively.

The position vectors of D, E, F are

Let G₁ be the point on AD dividing it internally in the ratio 2 : 1

Let G₂ be the point on BE dividing it internally in the ratio 2 : 1

Similarly if G₃ divides CF in the ratio 2 : 1 then

From (1), (2) and (3) we find that the position vectors of the three points G₁, G₂, G₃ are one and the same. Hence they are not different  points. Let the common point be denoted by G.

Therefore the three medians are concurrent and the point of concurrence is G.

5.If be the vectors represented by the three sides of a triangle, taken in order, then prove that 

Let ABC be a triangle such that

6.By using vectors, prove that a quadrilateral is a parallelogram if and only if diagonal bisect each other.

Let ABCD be a quadrilateral

First assume that ABCD is a parallelogram.

To prove that its diagonals bisect each other.

Let O be the origin of reference.

Since ABCD is a parallelogram,

That is, P. V of midpoint of BD = P. V of midpoint of AC. Thus the point, which bisects AC also bisects BD. Hence the diagonal of a parallelogram ABCD bisect each other.

Conversely suppose that ABCD is a quadrilateral such that  its diagonals bisects each other.

To prove that it is a parallelogram.

Let  be the position vectors of its vertices A, B, C and D respectively.

Since the diagonals AC and BD bisect each other, P. V of the midpoint of AC = P. V of the midpoint of BD.

Hence ABCD is a parallelogram.

7.Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it.

Let ABC be a triangle, and let O be the origin of reference.

Let D and E be the midpoints of AB and AC respectively.

8.Let be the position vectors of three distinct points A, B, C. If there exist scalars. l, m, n(not all zero) such that and l + m + n =0 then show that A, B and C lie on a line.

It is given that l, m, n are not all zero. So, let n be a non-zero scalar.

The point C divides the line joining A and B in the ratio m : l

Hence A, B and C lies on the same line.

9.Find the sum of the vectors and . Find also the magnitude of the sum.