Exam (elaborations)
Vector Algebra Questions and Answers
- Module
- MATHEMATICS VECTORS
- Institution
- The University Of Edinburgh (ED)
Vector Algebra Questions and Answers
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VECTOR PRODUCT OF FOUR VECTORS JEE MAINS - VOL - IVECTOR 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 ofa , 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 .
108
, 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
109
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