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Question

Mathematics Question on Vector Algebra

The value of [(a−b)(b−c)(c−a)][( a - b )( b - c )( c - a )] is equal to

A

11

B

22

C

00

D

2[abc]2[ a b c ]

Answer

00

Explanation

Solution

∴[(a−b)(b−c)(c−a)]\therefore[(a-b)(b-c)(c-a)]
=(a−b)⋅[(b−c)×(c−a)]=(a-b) \cdot[(b-c) \times(c-a)]
=(a−b)⋅[b×c−b×a−c×c+c×a]=(a-b) \cdot[b \times c-b \times a-c \times c+c \times a]
=(a−b)⋅[b×c−b×a+c×a]=(a-b) \cdot[b \times c-b \times a+c \times a]
[∵c×c=0][\because c \times c=0]
=a⋅[b×c]−a⋅[b×a]+a⋅[c×a]=a \cdot[b \times c]-a \cdot[b \times a]+a \cdot[c \times a]
−b⋅[b×c]+b⋅[b×a]−b⋅[c×a]-b \cdot[b \times c]+b \cdot[b \times a]-b \cdot[c \times a]
=[abc]−0+0+0−[bca]=[a b c]-0+0+0-[b c a]
=[abc]−[abc]=[a b c]-[a b c]
=0=0