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

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Question 20 If ๐‘Žย โƒ—, ๐‘ย โƒ—, ๐‘ย โƒ— are three vectors such that ๐‘Žย โƒ— + ๐‘ย โƒ— + ๐‘ย โƒ— = 0ย โƒ— , then prove that ๐‘Žย โƒ— ร— ๐‘ย โƒ— = ๐‘ย โƒ— ร— ๐‘ย โƒ— = ๐‘ย โƒ— ร— ๐‘Žย โƒ—, and hence show that [๐‘Žย โƒ—" " ๐‘ย โƒ—" " ๐‘ย โƒ— ] = 0. Theory Here [๐‘Žย โƒ—" " ๐‘ย โƒ—" " ๐‘ย โƒ— ] = ๐‘Žย โƒ—.(๐‘ย โƒ— ร— ๐‘ย โƒ— ) Given ๐‘Žย โƒ— + ๐‘ย โƒ— + ๐‘ย โƒ— = 0ย โƒ— ๐‘Žย โƒ—ร—(๐‘Žย โƒ—+๐‘ย โƒ—+๐‘ย โƒ— )= ๐‘Žย โƒ—ร—0ย โƒ— ๐‘Žย โƒ—ร—๐‘Žย โƒ—+๐‘Žย โƒ—ร—๐‘ย โƒ—+๐‘Žย โƒ—ร—๐‘ย โƒ—= 0ย โƒ— Since ๐‘Žย โƒ—ร—๐‘Žย โƒ—=0 " " 0+๐‘Žย โƒ—ร—๐‘ย โƒ—+๐‘Žย โƒ—ร—๐‘ย โƒ—=" " 0ย โƒ— ๐‘Žย โƒ—ร—๐‘ย โƒ—+๐‘Žย โƒ—ร—๐‘ย โƒ—=" " 0ย โƒ— ๐‘Žย โƒ—ร—๐‘ย โƒ—=โˆ’๐‘Žย โƒ—ร—๐‘ย โƒ— Since โˆ’๐‘Žย โƒ—ร—๐‘ย โƒ— = ๐‘ย โƒ—ร—๐‘Žย โƒ— ๐’‚ย โƒ—ร—๐’ƒย โƒ—=๐’„ย โƒ—ร—๐’‚ย โƒ— Similarly, ๐‘Žย โƒ— + ๐‘ย โƒ— + ๐‘ย โƒ— = 0ย โƒ— ๐‘ย โƒ—ร—(๐‘Žย โƒ—+๐‘ย โƒ—+๐‘ย โƒ— )= ๐‘ย โƒ—ร—0ย โƒ— ๐‘ย โƒ—ร—๐‘Žย โƒ—+๐‘ย โƒ—ร—๐‘ย โƒ—+๐‘ย โƒ—ร—๐‘ย โƒ—= 0ย โƒ— Since ๐‘ย โƒ—ร—๐‘ย โƒ—=0 ๐‘ย โƒ—ร—๐‘Žย โƒ—+0+๐‘ย โƒ—ร—๐‘ย โƒ—= 0ย โƒ— ๐‘ย โƒ—ร—๐‘Žย โƒ—+๐‘ย โƒ—ร—๐‘ย โƒ—=" " 0ย โƒ— ๐‘ย โƒ—ร—๐‘ย โƒ—=โˆ’๐‘ย โƒ—ร—๐‘Žย โƒ— ๐‘ย โƒ—ร—๐‘ย โƒ—=โˆ’๐‘ย โƒ—ร—๐‘Žย โƒ— Since โˆ’๐‘ย โƒ—ร—๐‘Žย โƒ— = ๐‘Žย โƒ—ร—๐‘ย โƒ— ๐‘ย โƒ—ร—๐‘ย โƒ—=๐‘Žย โƒ—ร—๐‘ย โƒ— Thus, ๐’‚ย โƒ—ร—๐’ƒย โƒ—=๐’„ย โƒ—ร—๐’‚ย โƒ— & ๐‘ย โƒ—ร—๐‘ย โƒ—=๐‘Žย โƒ—ร—๐‘ย โƒ— โˆด ๐’‚ย โƒ—ร—๐’ƒย โƒ—=๐’ƒย โƒ—ร—๐’„ย โƒ—=๐’„ย โƒ—ร—๐’‚ย โƒ— Now, we need to show that show that [๐‘Žย โƒ—" " ๐‘ย โƒ—" " ๐‘ย โƒ— ] = 0 [๐‘Žย โƒ— ๐‘ย โƒ— ๐‘ย โƒ— ]=๐‘Žย โƒ— . (๐‘ย โƒ—ร—๐‘ย โƒ— ) From (1): ๐‘ย โƒ—ร—๐‘ย โƒ— = ๐‘Žย โƒ—ร—๐‘ย โƒ— =๐‘Žย โƒ— . (๐‘Žย โƒ—ร—๐‘ย โƒ— ) Now, ๐‘Žย โƒ—ร—๐‘ย โƒ— will be a vector perpendicular to ๐‘Žย โƒ— And dot product of ๐‘Žย โƒ— with a vector perpendicular to ๐‘Žย โƒ— will be 0 as angle is 90ยฐ and cos 90ยฐ = 0 โˆด [๐‘Žย โƒ— ๐‘ย โƒ— ๐‘ย โƒ— ]=๐‘Žย โƒ— . (๐‘Žย โƒ—ร—๐‘ย โƒ— ) = 0 Hence proved

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