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If overset(to)(a) , overset(to)(b) " an...

If `overset(to)(a) , overset(to)(b) " and " overset(to)(c ) ` are three non- coplanar vectors then
`(overset(to)(a) + overset(to)(b) + overset(to)(c )) . [( overset(to)(a) + overset(to)(b)) xx (overset(to)(a) + overset(to)(c ))]` equals

A

0

B

`[ overset(to)(c)overset(to)(b) overset(to)c]`

C

`[overset(to)(c) overset(to)( b)overset(to)(c)]`

D

`[overset(to)(a) overset(to)(b)overset(to)(c)]`

Text Solution

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The correct Answer is:
D
`(vec(a) + vec(b + vec( c)).[(vec(a) + vec(b)) xx (vec(a) +vec( c))]`
`=(vec(a) + vec(b) + vec( c)).[vec(a) xx vec(a) +vec(a) xx vec( c) + vec( b) xx vec(a) +vec(b) xx vec(c )]`
`={vec(a).(vec(a) xx vec( c)) + vec(a) . (vec(b) xx vec(a)) + vec(a). (vec(b)xx vec(c )) }+ {vec(b). (|vec(a)xx vec(c )|) + vec(b). (vec(b) xx vec(a)) + vec(b). (vec(b) xx vec( c)) } +{vec(c ).(vec(a) xx vec( c)) + vec( c). (vec(b) xx vec(a)) + vec(c ). (vec(b) xx vec( c))}`
`= [vec(a) ,vec(b) ,vec(c )] + [vec(b) vec(a) vec(c )]+[vec(c ) vec(b) vec(a)] = [ vec(a) vec(b) vec(c)]`
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