Tiêu đề XII - maths - chapter 9 - VECTOR TRIPLE PRODUCT (237-246).pdf ✅

JEE-MAIN-JR-MATHS VOL-I VECTOR PRODUCT OF FOUR VECTORS NARAYANAGROUP 237  Vector triple Product : The vector product of a b and c is a vector triple Product of three vectors a,b and c . It is denoted by a b c  (a b )c  (a.c)b  (b.c)a . This is a vector in the plane of a and b .  a b c a c b a b c        . .    . This is a vector in the plane of b c,  ( ) ( ) a b c c a b        a,b ,c are non-zero vectors and ( ) ( ) & a b c a b c a c       are collinear (Parallel) (or) ( ) 0 a c b     Vector triple product is not associative. If a b c , , are non-zero, non-orthogonal vectors., then (a b )c  a (b c) .  a b c b c a c a b          ( ) ( ) 0   a (b c),b (c a),c (a b) are coplanar  i j k j k i k i j                0  i a i j a i k a  k   2a where a is any vector  a (b c )  b  (c  a)  (a b )c  2 [ ] [ ] a b b c c a a b c      Scalar Product of Four Vectors : ( ).( ) a b c d   is a scalar product of four vectors. It is a dot product of the vectors a b and c  d .  (a b ).(c  d )  a.c  (b.d )  a.d (b.c) b c b d a c a d . . . .     c b c d a b a d a c b d . . . . (  ).          2 2 2 2 a b a b a b a b a b       . .  Vector Product of Four Vectors : a b (c d ) is a vector product of four vectors.  a b c d a b d c a b c d                           c d a b c d b a                  a b c d b c d a c a d b a b d c             2     a b b c c a a b c          If a,b ,c are non coplanar vectors, i.e.,  a b c  0   then any vector r in space can be expressed as a linear combination of a,b ,c i.e., r b c r a b  r c a  r a b c a b c a b c a b c                        i.e., in the form r xa yb zc     If a,b ,c and d are coplanar then a b c d        0  If a,b ,c and d are parallel vectors (or) collinear vectors, then (a b )c d   0 W.E-1: Let a i j k b i j k       2 , 2 and a unit vector c be coplanar. If c is perpendicular to a , then c is equal to Sol: Required unit vector is     a a b a a b      a a b a b a a a b j k           . . 9 9      1 2      c j k VECTOR TRIPLE PRODUCT AND PRODUCT OF FOUR VECTORS SYNOPSIS
VECTOR PRODUCT OF FOUR VECTORS JEE-MAIN-JR-MATHS VOL-I 238 NARAYANAGROUP W.E-2: Let a i j and b i k     2 then point of intersection of the line r a b a and r b a b       is Sol: We have r a b a r b a          0      r b a r b a      r b a Similarly, the equation of the line r b a b    can be written as r a b    For the point of intersection of the above two lines, we have a b b a           1       r a b i j k 3 W.E-3:b c c a       is equal to Sol : b c c a           b c a c b c c a . .           a b c b c c c a        a b c c   VECTOR TRIPLE PRODUCT 1. If a i j k b i j k c i j k          , , 2 3 , then ( ) a b c    1) 2 6 2 i j k   2) 6 2 6 i j k   3)    6 2 6 i j k 4) 6 2 6 i j k   2. If a i j k b i j c i       , , and ( ) , a b c a b       then     1) 1 2) 0 3) -1 4) 2 3. a i j k b i j k       2 3 4 , , c i j k    4 2 3 then a b c      (EAM-2000) 1) 10 2) 1 3) 2 4) 5 4. a b c b c a c a b                1) 0 2) 0 3) 1 4) a b c  . 5. ( ) ( ) a b c a b c      if and only if 1) ( ) 0 a c b    2) a c b    ( ) 0 3) c b a    ( ) 0 4)   a b c  1   6. The vector ( ) a b c   is perpendicular to 1) c 2) a b  3) both 1 and 2 4) b c, 7. i a i j a j k a k          ( ) ( ) ( ) 1) 3a 2) 2a 3) a 4) 0 SCALAR PRODUCT OF FOUR VECTORS: 8. a i j k b i j k        2 3 , 2 4 , c i j k    , d i j k    then a b c d    . ___   1) 4 2) 24 3) 36 4) 4 9. If a b c b a b b c a c        . . .      then   1) 2 a 2) 2 b 3) 2 c 4) 0 10. a i b i a j b j            a k b k      1) a b. 2) 3 . a b  3) 0 4) 2 . a b  11. If a is parallel to b c  , then a b a c      1)   2 a b c. 2)   2 b a c. 3)   2 c a b. 4) 0 12. If a b, are two unit vectors such that a b   2 then the value of   a b a b    is 1) 1 2) 2 3) 4 4) 0 VECTOR PRODUCT OF FOUR VECTORS: 13. If a i j k b i j k       2 3 , 3 2 , c i j k d i j k       4 , 2 then a b c d        1) 24 2 i j k    2) 24i j k    3) 12 2 3  i j k    4)12 2 3 i j k    14 If four vectors a b c d , , , are coplanar, then ( ) ( ) a b c d    = 1)   a b c d   1)   b c d a   3)   c d a b   4) Null vector LEVEL - I (C.W)
JEE-MAIN-JR-MATHS VOL-I VECTOR PRODUCT OF FOUR VECTORS NARAYANAGROUP 239 15. If b c c a c        3 then   b c c a a b       1) 2 2) 7 3) 9 4) 11 LEVEL-I(C.W)-KEY 01) 2 02) 2 03) 4 04) 2 05) 1 06) 3 07) 2 08) 4 09) 2 10) 4 11) 1 12) 3 13) 1 14) 4 15) 3 LEVEL-I(C.W)-HINTS 1. a c b c b a . .     2. a c b a b            a c b b c a a b . .            b c c a . , .    3. a c b a b c . .     and 2 2 2 a i a j a k a a a 1 2 3 1 2 3      4.   a b c a c b a b c         . . 0     5. a b c a b c             c a b c b a a c b a b c . . . .           a b c c b a . . 0        a c b  0 6. Cross product of any two vectors is perpendicular to both the vectors 7.   i a i i i a i a i        . .       a a i i  .      3 2 a a a 8. . . . . a c a d b c b d 9. a c b b a b b c a b b c a c . . . . . . .                 2   a c b a c . .  2    b 10. Use scalar product of four vectors formula 11.      . . 2 . . . . . a a a c a b c b a a c b a b c       2    a b c b a c a . . . 0  12.     2   a b a b a b a b a b       .   13. Find     a b d c a b c d      14.     a b d c a b c d     0 0 0     15.   b c a c c  3       a b c 3   Required value 2   a b c  9   VECTOR TRIPLE PRODUCT 1. If a i j k b i j k       , , c i j k    then a b c      1) i j k   2) 2 2 i j  3) 3i j k   4) 2 2 i j k   2. If a i j k b i j k       2 3 , 2 and c i j k    3 2 , and a b c pi q j rk        , then p  q  r  a) -4 b) 4 c) 2 d) -2 3. If p q    (2, 10, 2), (3,1,2) and r  (2,1,3), then p q r    ( ) a) 2 b) 4 c) 0 d) 3 4. i j k j k i k i j          ( ) ( ) ( ) 1) i 2) j 3) k 4) Null vector 5. If a b c a b c         where a , b and c are any three vectors such that a b. 0  , b c. 0  then a and c are 1) Perpendicular 2) Parallel 3) Inclined at an angle of 3  between them 4) Inclined at an angle of 6  between tehm 6. The vectors a b c     is 1) Coplanar with b and c 2) Coplanar with a and b parallel to c 3) Coplanar with b and c , orthogonal to a 4) Coplanar with a and b , orthogonal to c 7. a i i a j j a k k               ___ 1) 2a 2) 2a 3) a 4) a LEVEL - I (H.W)
VECTOR PRODUCT OF FOUR VECTORS JEE-MAIN-JR-MATHS VOL-I 240 NARAYANAGROUP SCALAR PRODUCT OF FOUR VECTORS 8. If a i j k b i j k       , , c i j k d i j k       , then value of a b c d   .  is 1) 1 2) 0 3) -2 4) -1 9. a b c d a c b d K a d b c     . . . . .         then the value of K is 1) 1 2) 0 3) -2 4) -1 10. b c a d c a b d a b c d          . . .           1) 0 2) 1 3) 2 4) -1 11. a b c d b c a d     . . .      1) 0 2) 1 3) a c. 4) a c b d . .   12. If a , b lie in a plane normal to the plane containing c and d then a b c d    .  1) 4 2) 1 3) 0 4) 3 VECTOR PRODUCT OF FOUR VECTORS 13. a i j k b i j k c i j k           2 3 , 2 4 , , d i j k    3 4 2 . The value of a b c d        1)    70 28 64 i j k 2) 7 8 i j  3) 6 2 j k  4) i j k   4 14. a b c d lc md         then m is 1)   a b c d   2)   c b d   3)   b c d   4)    a b c   15. If    a b c    0 then a b b c c a            1) 0 2) A vector perpendicular to the plane of a ,b , c 3) A scalar quantity 4) 2   a b c   LEVEL-I(H.W)-KEY 01) 2 02) 1 03) 3 04) 4 05) 2 06) 3 07) 2 08) 2 09) 4 10) 1 11) 4 12) 3 13) 1 14) 4 15) 1 LEVEL-I(H.W)-HINTS 1. a c b a b c . .     2. a c b a b c . .     3.  p r q p q r . .     4. a b c a c b a b c     ( ) ( . ) ( . ) 5. c a b c b a a c b a b c . . . .             c b a a b c . .     a c, are parallel 6. Cross product of any two vectors is perpendicular to both the vectors and a b c c a b c b a       . .    7.   a i i i a i             i i a i a i . .        a i a i  .        3 2 a a a 8. . . . . a c a d b c b d 9. a c b d a d b c a c b d K a d b c . . . . . . . .                 K 1 10. Use scalar product of four vectors formula 11.    . . . . . . a c a d b c a d b c b d  12. a b c d a b c d            . 0   13. a b i j k      10 9 7 c d i j k     6 7 a b c d i j k           70 28 64 14.         a b d c a b c d lc md        m a b c    