**Solutions of Vector Geometry for SEE/SLC Exam**

Q. no. 1.In the given figure, AB, BE and CF are the medians of the triangle ABC, prove that:

vector AD + vector BE + vector CF = 0

**2. In the adjoining figure, RM and QN are the medians of the triangle ΔPQR. If RM = QN, prove by vector method that triangle ΔPQR is an isosceles triangle.**

**3. ABCD is a parallelogram and G is the point of intersection of its diagonals. If O is any point prove that: vector OA + vector OB + vector OC + vector OD = 4 times vector OG**

**4. In the given figure, PQRS is a parallelogram. M and N are two points on the diagonal SQ. If SM = NQ, prove by vector method that PMRN is a parallelogram.**

**5. Prove by vector method that the quadrilateral whose diagonals bisect each other is a parallelogram.**

**6. In the given figure, P, Q, R and S are the mid - points of the sides AB, AD, CD and BC respectively of the quadrilateral ABCD. Prove by vector method that PR and QS bisect each other.**

**7. Prove vectorially that a parallelogram with equal diagonals is a rectangle.**

**8. ABCD is a quadrilateral. Show that vector AB + vector AD + vector CB + vector CD = 4 times vector PQ where P and Q are the mid - points of AC and BD respectively.**

**9. If the diagonals of a parallelogram MNPQ bisect each other at right angle, prove vectorially that the parallelogram MNPQ is a rhombus.**

**10. In the given figure, ABCD is a parallelogram. L and M are the mid - points of DC and CB respectively. Prove that 2 (vector AL + vector AM) = 3 times vector AC**

**11. In the adjoining figure, PQRS is a trapezium where PQ//RS; X and Y are the mid - points of PS and QR respectively. Prove vectorially that: vector XY = 1/2 ( vector PQ + vector SR)**

**12. If a line is drawn from the centre of a circle to the mid - point of a chord, prove by vector method that the line is perpendicular to the chord.**

**13. The position vectors of A and B are (1,1) and (4,1) respectively. Find the ratio in which P(2,1) divides AB internally.**

**14. In the given figure, if vector DE = vector AD + vector DB + vector DC, prove that ABEC is a parallelogram.**

**15. Find the position vector of a point on the y-axis which divides the line joining the points (1,4) and (-2,-5) in the ratio 1:2.**

**16. In the parallelogram ABCD if G is the mid - point of the intersection of its diagonals, prove that:**vector OA + vector OB + vector OC + vector OD = 4 times vector OG.

**17. If vector OA = 4 times vector i + 4 times vector j, vector OB = 7 times vector i + 7 times vector j, vector OP = －vector i － vector j, in what ratio does P divide AB externally?**

**18. Prove by vector method that the medians of an equilateral triangle are equal.**

**19. ABCDE is a pentagon, prove that:**(i) vector AB + vector BC + vector CD + vector DE + vector EA = 0

(ii) vector AB + vector AE + vector BC + vector DC + vector ED + vector AC = 3 times vector AC.

**20. If D, E and F are the mid - points of the sides of triangle ΔABC, prove that: vector OA + vector OB + vector OC = vector OD + vector OE + vector OF.**

**21. In the trapezium ABCD, AB//DC, P and Q are the mid - points of of BD and AC respectively. Prove that: vector PQ = 1/2 (vector DC － vector AB)**

**22. In the parallelogram ABCD, the diagonals AC and BD intersect at P. Prove that: vector PA + vector PB + vector PC + vector PD =0**

**23. If AC and BD are two diagonals of a parallelogram ABCD, prove that:**(i) AC² + BD² = 2 (AB² + AD²)

**(ii) AC**

**²**

**－ BD**

**²**

**= 4 times vector AB * vector AD**

**24. If A, B, C and D are any four points then prove that: vector AB * vector CD + vector BC * vector AD + vector CA * vector BD = 0**

**25. In regular hexagon ABCDEF, prove that: vector AB + vector AC + vector AD + vector EA + vector FA = 4 times vector AB**

**26. If ABCDEF is a regular hexagon, prove that:**(i) vector AB + vector DE = 0

(ii) vector AB + vector BC + vector CD = 2 times vector BC

(iii) vector AB + vector AC + vector AD + vector AE + vector AF = 6 times vector AO

**27. In the given figure, AB is the diameter of the circle and CD⟂AB then prove that:**

**(i) CD**

**²**

**= vector AD * vector DB**(ii) vector AC = vector AD * vector AB