CBSE Class 12 Sample Paper for 2018 Boards

Class 12
Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

### If a, b, c are three vectors such that a + b + c = 0 ,Β  then prove that a Γ b = b Γ c = c Γ a, and hence show that [a b c] = 0.

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

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Question 20 If π β, π β, π β are three vectors such that π β + π β + π β = 0 β , then prove that π β Γ π β = π β Γ π β = π β Γ π β, and hence show that [π β" " π β" " π β ] = 0. Theory Here [π β" " π β" " π β ] = π β.(π β Γ π β ) Given π β + π β + π β = 0 β π βΓ(π β+π β+π β )= π βΓ0 β π βΓπ β+π βΓπ β+π βΓπ β= 0 β Since π βΓπ β=0 " " 0+π βΓπ β+π βΓπ β=" " 0 β π βΓπ β+π βΓπ β=" " 0 β π βΓπ β=βπ βΓπ β Since βπ βΓπ β = π βΓπ β π βΓπ β=π βΓπ β Similarly, π β + π β + π β = 0 β π βΓ(π β+π β+π β )= π βΓ0 β π βΓπ β+π βΓπ β+π βΓπ β= 0 β Since π βΓπ β=0 π βΓπ β+0+π βΓπ β= 0 β π βΓπ β+π βΓπ β=" " 0 β π βΓπ β=βπ βΓπ β π βΓπ β=βπ βΓπ β Since βπ βΓπ β = π βΓπ β π βΓπ β=π βΓπ β Thus, π βΓπ β=π βΓπ β & π βΓπ β=π βΓπ β β΄ π βΓπ β=π βΓπ β=π βΓπ β Now, we need to show that show that [π β" " π β" " π β ] = 0 [π β π β π β ]=π β . (π βΓπ β ) From (1): π βΓπ β = π βΓπ β =π β . (π βΓπ β ) Now, π βΓπ β will be a vector perpendicular to π β And dot product of π β with a vector perpendicular to π β will be 0 as angle is 90Β° and cos 90Β° = 0 β΄ [π β π β π β ]=π β . (π βΓπ β ) = 0 Hence proved