Xii Vector Algebra Assignment

ASSIGNMENT CLASS XII VECTOR ALGEBRA             1. In a regular hexagon ABCDEF, if AB  a and BC  b , then express CD, DE, EF , FA, AC , AD, AE and CF   in terms of a and b .       2. If a  i  j, b  j  k , c  k  i , find a unit vector in the direction of a  b  c .

3. The position vectors of the points P, Q and R are i  2 j  3 k ,  2 i  3 j  5 k , 7 i  k respectively. Prove that P, Q and R are collinear.   4. If a  i  2 j  3 k , b  2 i  4 j  5 k represents two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram.

5. Prove that the points i  j , 4 i  3 j  k , 2 i  4 j  5k are the vertices of a right angled triangle. 6. If the position vectors of the vertices of a triangle ABC are i  2 j  3k , 2 i  3 j  k , 3 i  j  2k , prove that  ABC is an equilateral triangle. 7. Write the position vector of a point dividing the line segment joining points A and B with position vectors       a and b externally in the ratio 1: 4, where a  2 i  3 j  4k and b   i  j  k .       8. Find the projection of b  c on a , where a  2 i  2 j  k , b  i  2 j  2 k and c  2 i  j  4k .       9. If a  i  j  2k and b  3 i  2 j  k , find the value of a  3 b . 2a  b .

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 10. Find a vector whose magnitude is 3 units and which is perpendicular to each of the vectors a  3 i  j  4k  and b  6 i  5 j  2k .            11. If a , b and c be three vectors such that a  b  c  0 and a  3, b  5, c  7 , find angle between a and b .           12. If a and b are vectors such that a  2, b  3 and a . b  4, find a  b and a  b .

 1 13. If a and b are unit vectors and  is the angle between them, prove that sin   1 a  b and cos  a  b . 2 2 2 2 14. Show that the points A , B and C with position vectors 2i  j  k , i  3 j  5k , 3 i  4 j  4k respectively, are the vertices of the right triangle. Also, find the remaining angles of the triangle.       15. If a  i  2 j  3k and b  3 i  j  2k , then show that a  b is perpendicular to a  b .       16. Find the angle between the vectors a  b and a  b , if a  2 i  j  3k and b  3 i  j  2k .   17. Express the vectors a  5 i  2 j  5k as sum of two vectors such that one is parallel to the vector b  3 i  k  and the other is perpendicular to b .

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18. The dot products of a vector with the vectors i  j  3k , i  3 j  2k and 2 i  j  4k are 0, 5, 8 respectively. Find the vector.   19. Find a unit vector perpendicular to each of the vectors a  4 i  j  3k and b  2 i  2 j  k .

      20. If a  26, b  7 and a  b  35, find a . b .  21. Find the area of the triangle whose adjacent sides are determined by the vectors a   2 i  5k and  b  i  2 j  k .  22. Find the area of the parallelogram whose adjacent sides are determined by the vectors a  3 i  j  2k and  b  i  3 j  4 k .  23. Find the area of the parallelogram whose diagonals are determined by the vectors a  2 i  3 j  6k and  b  3 i  4 j  kˆ .    24. Show that points whose position vectors are a  5 i  6 j  7k , b  7 i  8 j  9k , c  3 i  20 j  5k are collinear.         25. Let a  i  j , b  3 j  k , c  7 i  k . Find a vector d such that it is perpendicular to both a and b ,and c . d  1    26. If a , b , c are the position vectors of the vertices A, B and C of a  ABC respectively, find an expression for the area of  ABC and hence deduce the condition for the points A, B and C to be collinear.         27. If a  i  j  k , c  j  k are given vectors, then find a vector b satisfying equations a  b  c and a . b  3 .             28. If a , b , c are three vectors such that a  b  c  0 , then prove that a  b  b  c  c  a .                 29. If a  b  c  d and a  c  b  d , show that a  d is parallel to b  c , where a  d , b  c .             30. If a . b  a . c and a  b  a  c and a  0 , then show that b  c .

ANSWERS                     1. CD  b  a , DE   a , EF   b , FA  a  b , AC  a  b , AD  2b , AE  2b  a , CF   2 a 1   1  1  11 i j k i  2 j  8 k 2. 4. 3 i  6 j  2 k , 7. 3 i  j  5k 8. 2 9. 15 7 3 3 69  35   6   1 10. 2 i  2 j  k 11. 600 12. 5 , 21 14. cos1  16.  , cos   2  41   41  1  1 17. 6i  2k ,  i  2 j  k 18. i  2 j  k 19. i  2 j  2k 20. 7 21. 165 sq.units 3 2 1       1 1   22. 10 3 sq.units 23. 1274 sq.units 25. i  j  3k 26. ar   ABC   a  b  b  c  c  a ; 2 4 2        1 a b  b c  c  a  0 27. b  5 i  2 j  2k 3

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