Vector Algebra Questions and Answers

VECTOR PRODUCT OF FOUR VECTORS JEE MAINS - VOL - I

VECTOR TRIPLE PRODUCT AND PRODUCT OF FOUR VECTORS

SYNOPSIS a .b a .d
 (a  c ).b  d  
 Vector triple Product : c .b c .d
The vector product of a  b and c is a vector 2 2 2 2

triple Product of three vectors a, b and c . It is
  a  b  . a  b   a  b  a b   a .b 
 Vector Product of Four Vectors :
denoted by a  b  c
a  b  (c  d ) is a vector product of four
 (a  b )  c  (a .c )b  (b .c )a . This is a vector vectors.
in the plane of a and b .
 a b c  d    a b d  c   a b c  d
 a   b  c    a .c  b   a .b  c . This is a vector
  c d a  b   c d b  a
in the plane of b , c
  a b c  d  b c d  a   c a d  b   a b d  c
 (a  b )  c  c  (a  b )
2
 a , b , c are non-zero vectors and   a  b b  c c  a    a b c 

( a  b )  c  a  (b  c )  a & c are collinear  If a , b , c are non coplanar vectors,
(Parallel) (or) ( a  c )  b  0 i.e.,  a b c   0 then any vector r in space can
 Vector triple product is not associative. If a , b , c be expressed as a linear combination ofa , b , c
are non-zero, non-orthogonal vectors., then
 r b c   r c a  b   r a b  c
(a  b )  c  a  (b  c ) . r  a 
i.e.,  a b c   a b c   a b c 
 a  (b  c )  b   c  a   c  (a  b )  0
i.e., in the form r  xa  yb  zc
 a  (b  c ), b  (c  a ), c  (a  b ) are coplanar
i   j  k   j  k  i   k  i  j   0
 If a , b , c and d are coplanar then

 i  a  i   j  a  i   k  a  k   2a
a  b   c  d   0
where a is any vector  If a , b , c and d are parallel vectors (or) collinear

 a  (b  c )  b  ( c  a )  ( a  b )  c vectors, then (a  b )  c  d   0
 To find the direction of a line with greatest slope:
 [a  b b  c c  a ]  [a b c ]2
 Scalar Product of Four Vectors : Let  1 ,  2 be two planes intersecting in a line l1
then the line of greatest slope in 1 is the line lying
( a  b ).(c  d ) is a scalar product of four vectors.
in the plane 1 and perpendicular to the line l1 .
It is a dot product of the vectors a  b and c  d .
Note: Let a , b be the vectors along the normals to the
 (a  b ).(c  d )  a .c  (b .d )  a .d  (b .c )
planes 1 and  2 respectively then the vector
a .c a .d

b .c b .d
 
a  a  b will be along the line of greatest slope
in 1 .

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, JEE MAINS - VOL - I VECTOR PRODUCT OF FOUR VECTORS

W.E-1: Let a  2 i  j  k , b  i  2 j  k and a 1) 1 2) 0 3) -1 4) 2
unit vector c be coplanar. If c is 3. a  2 i  3 j  4k , b  i  j  k ,
perpendicular to a , then c is equal to
c  4 i  2 j  3k then a   b  c  
a  a  b  (EAM-

Sol: Required unit vector is a  a  b
  2000)
1) 10 2) 1 3) 2 4) 5
a   a  b    a .b  a   a .a  b  9 j  9k
4. a  b  c   b   c  a   c   a  b  
1
c  
2
 j  k  1) 0 2) 0 3) 1 4)  a  b  .c

W.E-2: Let a  i  j and b  2 i  k then point of 5. (a  b )  c  a  (b  c ) if and only if
intersection of the line 1) (a  c )  b  0 2) a  (c  b )  0
r  a  b  a and r  b  a  b is 3) c  (b  a )  0 4)  a b c   1
Sol: We have r  a  b  a   r  b   a  0 6. The vector (a  b )  c is perpendicular to

 r  b  a  r  b  a 1) c 2) a  b 3) both 1 and 2 4) b , c
 r  b  a 7. i  (a  i )  j  (a  j )  k  (a  k ) 
Similarly, the equation of the line r  b  a  b 1) 3a 2) 2a 3) a 4) 0
can be written as r  a   b Scalar product of four vectors:
For the point of intersection of the above two lines, 8. a  2 i  3 j  k , b   i  2 j  4k ,
we have a   b  b   a      1 c  i  j  k , d  i  j  k then
 r  a  b  3i  j  k
 a  b  .  c  d   ___
W.E-3:  b  c    c  a  is equal to 1) 4 2) 24 3) 36 4) 4

  
Sol :  b  c    c  a    b  c  .a c   b  c  .c a  9. If  a  b  c  b    a . b  b . c     a . c 
then  
  a b c  c   b c c  a   a b c  c 2
2 2
1) a 2) b 3) c 4) 0

LEVEL - I (C.W) 10.  a  i  b  i    a  j  b  j  
Vector triple product:  a  k  b  k  
1. If a  i  j  k , b  i  j  k , c  2 i  3 j  k , 1) a .b 2) 3  a .b  3) 0 4) 2  a .b 
then (a  b )  c 
11. If a is parallel to b  c , then  a  b   a  c  
1) 2 i  6 j  2k 2) 6i  2 j  6k
2 2

3) 6i  2 j  6k 4) 6i  2 j  6k
1) a  b .c  2) b  a .c 
2
2. If a  i  j  k , b  i  j , c  i and 3) c  a .b  4) 0
(a  b )  c   a   b, then   
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