Today is our topic of discussion Planar vector exercises-12.
Planar vector exercises-12
1. if AB || DC then
(i) AB =m. DC where m is a scalar quantity
(ii) AB = DC
(iii) AB = CD
Which one of the above sentences is true?
1) i
3) i and ii
2) ii
4) i, ii and iii
2. If the two vectors are parallel
(i) Parallelogram law is applicable in case of their addition
(ii) Triangle law is applicable in case of their addition
(iii) Their lengths are always equal
Which one is true among the above sentences?
1) i
2) ii
3) i and ii
4) i, ii and iii
3. Which one of the following is true if AB = CD and AB ||CD?
1) AB = CD
2) AB =m. CD where m > 1
3) AB + DC < 0
4) AB +m CD = 0 . where m > 1
Answer the questions 4 and 5 on the basis of the information given below:
C is any point on the line segment AB and a b and c are respectively the position vectors of the points A, B and C with respect to a vector origin.
4. vec AA vector is a –
(i) Point vector
(ii) Unit vector
(iii) Zero vector
Which one of these is right?
1) i, ii
2) i, iii
3) ii, iii
4) i, ii, and iii
5. Which one is true in case of AABC?
1) AB + BC = CA
2) AB+ AC = BC
3) CB+BA+CA = 0
4) AB + BC + CA = 0
6. If the diagonals of parallelogram ABCD are AC and BD then, express AB and AC through AD and BD and show that, AC BD = 2BC and + AC – BD = 2AB.
7. Show that,
1)(a+b)-a-b
2) if a+b=c then a = c-b
8. Show that,
1) a + a = 2a
2) (m-n)ama – na
3) m(a – b) ma – mb
9. Show that,
1) If each of a, b is a nonzero vector, then amb can be true if and only if, a is parallel to b.
2) If both a, b are nonzero and non-parallel vectors, and if ma + nb = 0 then show that, m = n = 0.
10. If a, b, c, d are the position vectors respectively of the points A, B, C, D then show that, ABCD will be a parallelogram if and only if b-a=c-d.
11. Prove with the help of vectors that the straight line drawn from the middle point of a side of a triangle and parallel to another side passes through the middle point of the third side.
12. If the diagonals of a quadrilateral bisect each other, prove that it is a parallelogram.
13. Prove with the help of vectors that the straight line joining the middle points of the non-parallel sides of a trapezium is parallel to and half of the sum of the parallel sides.
14. Prove with the help of vectors that the straight line joining the middle points of the diagonals of a trapezium is parallel to and half of the difference of the parallel sides.
15. D and E are respectively the middle points of the sides AB and AC of the triangle ABC.
1) Express (AD + DE) in terms of AC
2) Prove with the help of vectors that, BC || DE and DE = BC
3) If M and N are the middle points of the diagonals of the trapezium BCED, then prove with the help of vectors that MN || DE || BC and 1 MN= (BC – DE) 2
16. D, E and F are the middle points of the sides BC, CA and AB of the ΔABC, respectively.
1) Express AB in terms of the vectors BE and CF.
2) Prove that AD + BE+CF = 0
3) Prove with the help of vectors that the straight line drawn through F parallel to BC must go through E.
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