vector

Vectors 379
EXERCISE 5.2
1. Determine such that
^ ^ ^ ^ ^ ^ ^ ^
a i j k, b 2i 4 k, and c i j 3 k are coplanar. Ans. = 5/3
2. Show that the four points
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
6 i 3 j 2 k,3i 2 j 4 k,5i 7 j 3 k and 13 i 17 j k are coplanar.
3. Find the constant a such that the vectors
^ ^ ^ ^ ^ ^ ^ ^ ^
2 i j k, i 2 j 3 k,and 3 i a j 5 k are coplanar. Ans. – 4
4. Prove that four points
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
4 i 5 j k, ( j k), 3 i 9 j 4 k,4( i j k)
are coplanar.
5. If the vectors a, b and c are coplanar, show that
a b c
a . a a . b a . c
b . a b . b b . c
5.19 VECTOR PRODUCT OF THREE VECTORS (A.M.I.E.T.E., Summer, 2004, 2000)
= 0
Let a , b and c be three vectors then their vector product is written as a ( b ×
c ).
Let
^ ^ ^
a = a i a j a k
1 2 3 ,
^ ^ ^
b = b1 i b2 j b3 k,
^ ^ ^
c = c1 i c2 j c3
k
^ ^ ^ ^ ^ ^ ^ ^ ^
( a1 i a2 j a3 k) ( b1 i b2 j b3 k) ( c1 i c2 j c3
k)
^ ^ ^ ^ ^ ^
= a1 i a2 j a3 k bc 2 3 bc 3 2 i b3c1 b1c3 j bc 1 2
bc 2 1 k
a ( b c ) =
=
( ) [( ) ( ) ( ) ]
^
2 1 2 2 1 3 3 1 1 3 3 2 3 3 2 1 1 2
2 1
[ a ( b c b c ) a ( bc bc )] i [ a ( bc bc ) a ( b c b c )] j
[ a ( b c b c ) a ( bc bc) k]
1 3 1 1 3 2 2 3 3 2
^ ^ ^ ^ ^ ^
= ac 1 1 ac 2 2 ac 3 3 b1 i b2 j b3 k a1 b1 ab 2 2 ab 3 3 c1 i c2 j c3
k
=
( )( ) ( )( )
( a . c) b ( a . b) c .
Ans.
Example 8. Prove that :
a ( b c) b ( c a) c ( a b ) 0 (Nagpur University, Winter 2008)
Solution. Here, we have
a ( b c) b ( c a) c ( a b )
=
[( a . c) b ( a . b) c] [( b . a) c ( b . c) a] [( c . b) b ( c . a) b]
=
[( b . a) c ( a . b) c] [( c . b) a ( b . c) a] [( a . c) b ( c . a) b]
= [( a . b) c ( a . b) c] [( b . c) a ( b . c) a] [( c . a) b ( c . a) b]
= 0 + 0 + 0 = 0 Proved.
Example 9. Prove that :
^ ^ ^ ^ ^ ^
i ( a i) j ( a j) k ( a k) 2 a (Nagpur University, Winter 2003)
^ ^ ^
Solution. Let a = a i a j a k
1 2 3
^
^