If a, b, c are non-coplanar vectors then Statement-1: b×c,c×a,a×b are non-coplanar.Statement-2: [b×c c×a a×b]=2[a b c]2

[b×c c×a a×b]=((b×c)×(c×a))⋅(a×b)=((b⋅(c×a))c−(c.(c×a)b)⋅(a×b)=[a⋅b⋅c]c⋅(a×b)(c⋅(c×a)=0)=[abc][abc]=[a b c]2Since R.H.S. is non-zero so is L.H.S. Hence b × c, c × a, a × b are non-coplanar.

If a, b, c are non-coplanar vectors then Statement-1: b×c,c×a,a×b are non-coplanar.Statement-2: [b×c c×a a×b]=2[a b c]2

[b×c c×a a×b]=((b×c)×(c×a))⋅(a×b)=((b⋅(c×a))c−(c.(c×a)b)⋅(a×b)=[a⋅b⋅c]c⋅(a×b)(c⋅(c×a)=0)=[abc][abc]=[a b c]2Since R.H.S. is non-zero so is L.H.S. Hence b × c, c × a, a × b are non-coplanar.

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If $a$ , $b$ , $c$ are non-coplanar vectors then
Statement-1: $b \times c, c \times a, a \times b$ are non-coplanar.
Statement-2: $[b \times c c \times a a \times b] = 2 [a b c]^{2}$

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STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is False, STATEMENT-2 is True

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is NOT a correct explanation for STATEMENT-1

answer is C.

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Detailed Solution

$\begin{array}{l} [b \times c c \times a a \times b] \\ = ((b \times c) \times (c \times a)) \cdot (a \times b) \\ = ((b \cdot (c \times a)) c - (c . (c \times a) b) \cdot (a \times b) \\ = [a \cdot b \cdot c] c \cdot (a \times b) (c \cdot (c \times a) = 0) \\ = [abc] [abc] = [a b c]^{2} \end{array}$