MATHEMATICS FOR XII

Vector Made Easy By Rajesh Kumar Giri

During my college life i faced a lot of problems to solve the questions in vector. I tried to get its solution from my tutor, professor and several help books but i failed. I went through concepts, illustrations, examples and work assignments. I was totally confused but being a struggler find out the best way to solve vector problems and decided to share you. Now it is your turn to rate me after observing the fact. SPECIAL FEATURES *CLEAR-CUT CONCEPTS *STEP BY STEP SOLUTION *COVERING CBSE LATEST SYLLABUS *WORKING ASSIGNMENTS *TOPIC-WISE ILLUSTRATION Contents of the E-book 1. DETERMINATION OF MAGNITUDE OF VECTORS 2. ADDITION OF VECTOR 3. FINDING OUT SCALAR AND VECTOR COMPONENTS 4. FINDING OUT UNIT VECTOR 5. TO PROVE COLLINIARITY OF VECTORS

CHAPTER-WISE ASSIGNMENTS FOR CBSE XII (All Pages are being developed So please subscribe the blog to keep yourself updated or call me- Rajesh Kumar Giri (91-8527307169)

• RELATION AND FUNCTIONS
• INVERSE TRIGONOMETRICAL FUNCTION
• MATRICES
• DETERMINANTS
• CONTINUITY AND DIFFERENTIABILITY
• APPLICATION OF DERIVATIVES
• INTEGRALS
• APPLICATION OF INTEGRALS
• DIFFERENTIAL EQUATIONS
• VECTOR ALGEBRA
• THREE DIMENSIONAL GEOMETRY
• LINEAR PROGRAMMING
• PROBABILITY

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VECTORS-AN INTRODUCTION-1

BASIC LEVEL-A(INTRODUCTION TO VECTORS)

1.For what values of ‘?’, the vectors (2î - 3 j )  and  (?î - 6j) are parallel ?

2.Find the position vector of the centroid of a ?ABC where a  , b  and c  are the position vectors of the vertices A, B and C respectively.

3.Find position vectors of the points which divides the join of the points 2a  - 3b   and 3a  -  2b   externally   in the ratio 2:3.

4.Find the projection of the vector   î +3 j +7k   on  the vector   7î – j + 8k.

5.Find the area of the parallelogram whose diagonals are  2î - 3j + 4k  and  -3î + 4j – k

6.Find the cosine of an acute angle between the vectors  2î - 3j + k  and   î + j -2 k

7.If  | a + b | =  | a - b |  then find the angle between  a   and  b  .

8.If  | a  | = 5,  | b | = 13  and  | a  x  b  | = 25. Find  a . b

9.Find the position vector of the mid point of the vector joining the points  P(2,3,4)  and  Q(4,1,-2)

10.Find the value of   ‘x’   for which  x(î + j + k ) is a unit vector.

11.If the position vector  a    of the point (5,n) is such that  | a  | = 13, find the value of n

12.If the vector  a   = 2î - 3j  and  b  = -6î + mj  are collinear , find the value of m

13.If a vector makes angles  a, ß, ?  with x-axis, y-axis, z-axis respectively, then what is the value of     sin2a + sin2ß + sin2? .

14.If  | a  + b |2   =  | a  |2  +  | b | 2 ,  what is the angle between  a    and  b  ?

15.If  a   is a unit vector and  ( x  - a ).( x + a ) = 8 , then find  | x |

16.If a unit vector  a   makes angles  p with î ,  p  with  j  and an acute angle ? with k ,  then     3              4  find ‘?’                                                                                                                                                                         and hence the components of  a

17.Let a  , b  and c   be the three vectors such that | a  | = 3, | b | = 4, | c  | = 5and each one of them being   perpendicular to the sum of the other two, find |a + b+ c  |

18.If with reference to right handed system of mutually perpendicular unit vectors î, j and k, a  =  3î – j, ß   =  2î + j-3 k, then express ß  in the form of  ß = ß1 + ß2  where ß1 is parallel to a  and  ß2  is perpendicular to a

19.Let  a   = î + 4j +2k ,  b  = 3î - 2j + 7k  and  c  =  2î – j + 4k . Find a vector d   which is perpendicular to     both  a  and  b   and  c . d  = 15

20.If a , b and c are the position vectors of the vertices A,B and C of a ?ABC. Show that the area of the ?ABC is 1 |a  x b +  b x c + c  x a  |. Also find the condition of the    2collinearity of these points.

21. If a , b , and c  represents the vectors  BC,  CA  and  AB of a ?ABC, then show that                                 a  x  b =  b  x  c =    c  x  a     Hence deduce sine formula for a triangle.

22.If a , b and  c  are vectors such that  a . b = a . c  , a x b = a x c  and  |a | = 0 then prove that  b = c

23.Three vertices of a triangle are A(0,-1,-2) , B(3,1,4) and  C(5,7,1). Show that it is a right angled   triangle.    Also find the other two angles

24.If   a  x  b  =  c  x d    and    a  x  c   =  b  x  d  ,  show that    a  -  d   is parallel to   b  -  c  where a = d  and b ? c.

25.If  a , b , and c   are three mutually perpendicular vectors of equal magnitude, prove that  a + b +c   is equally inclined with vectors  a , b and  c

 

PREPARED BY- RAJESH KUMAR GIRI

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