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If the vectors `overset(to)(b), overset(to)(c ) , overset(to)(d)` are not coplanar then prove than the vectors `(overset(to)(a) xx overset(to)(b)) xx (overset(to)(c ) xx overset(to)(d)) + (overset(to)(a) xx overset(to)(c )) xx (overset(to)(d) xx overset(to)(b))`
`+(overset(to)(a) xx overset(to)(d)) xx (overset(to)(b) xx overset(to)( c))` is parallel to `overset(to)(a) `

Text Solution

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Considering first part `(vec(a) xx vec(b)) xx (vec(c ) xx vec(d))`
Let `vec(c ) xx vec(d) = vec( e)`
`(vec(a) xx vec(b)) xx vec(e ) = (vec(a )"." vec(e )) vec(b) - (vec(b) "." vec(e )) vec(a)`
`[:' (vec(a) xx vec(b)) xx vec( c) = (vec(a) xx vec(c ) ) vec(b) - (vec(b)". " vec(c )) vec(a)]`
`={vec(a) " ." (vec(c ) xx vec( d))} vec(b) -{vec(b).(vec(c ) xx vec(d))}vec(d)`
` = [vec(a) vec(c) vec(d)]vec(b) -[vec(b) vec(c ) vec(d)]vec(a)`
Similarly
`(vec(a) xx vec(c )) xx (vec(d) xx vec(b)) = [ vec(a) vec(d) vec(b)] vec( c) - [vec(c) vec(d) vec(b)] vec(a)`
` =[vec(a) vec(d) vec(b)]vec(c )- [vec(b) vec( c) vec(d )]vec(a)`
Also `(vec(a) xx vec(d)) xx (vec(b) xx vec(c )) =- (vec(b) xx vec(c )) xx (vec(a) xx vec(d))`
`=(vec(b) xx vec(c )) xx (vec(d) xx vec(a)) =[vec(b) vec(d) vec(a)] vec(c ) -[vec( c)vec(d) vec(a) ]vec(b)`
` =[vec(a) vec(d)vec(b)]vec(c) -[vec(a)vec(c) vec(d)]vec(b)`
From Eq . (i) (ii) and (iii)
`(vec(a) xx vec(b)) xx (vec(c ) xx vec(d)) + (vec(a ) xx vec(d)) xx (vec(d) xx vec(b)) + (vec(a) xx vec(d)) xx (vec(b) xx vec(c))`
`=[vec(a) vec(c)vec(d)]vec(b) -[vec(b)vec(c )vec(d)]vec(a)+[vec(a) vec(d)vec(b) vec(c)-[vec(b)vec(c)vec(d)]vec(a)-[vec(a)vec(d)vec(b)]vec(c)`
` -[vec(a)vec(c)vec(d)]vec(b) =2[vec(b) vec(c )vec(d)]vec(a)`
`:. ` Parallel to `vec(a)`
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