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Question : 21 of 101
Marks:
+1,
-0
Solution:
CONCEPT: Properties of Scalar Triple Product -
[abc]=[bca]=[cab] -
[abc]=−[bac]=−[cba]=−[acb] -
[(a+b)cd]=[acd]+[bcd] -
[λa,bc]=λ[abc] - Three non-zero vectors
, and
are coplanar if and only if
[abc]=0 CALCULATION: Given:
, and
are coplanar i.e
[]=0 ⇒(2a×3)⋅4=[2a4c]‌ and ‌(5×3c]⋅6a=[536a] ⇒(2a×3)⋅4+(5b×3c)⋅6=[2a34c]+[536a] As we know that,
[abc]=[bca]=[cab] ⇒(2×3)⋅4+(5×3c]b⋅6=[234c]+[653c] As we know that,
[λa,bc]=λ[abc] ⇒[234c]+[653c]=24[c]+90[a] As we know that, vectors
, and
are coplanar if and only if
[abc]=0 ⇒[234c]+[6a53c]=0 ⇒(2×3)⋅4+(5×3c]⋅6=0 Hence, correct option is 3 .
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