Question

If the vectors `overset(to)(a) , overset(to)(b) " and " overset(to)( c)` from the sides BC,CA and AB respectively of a`DeltaABC` then
A. `overset(to)(a),overset(to)(b)+overset(to)(b).overset(to)(c)+overset(to)(c).overset(to)(a)=0`
B. `overset(to)(a)xxoverset(to)(b)=overset(to)(b)xxoverset(to)(c)=overset(to)(c)xxoverset(to)(a)`
C. `overset(to)(a).overset(to)(b)=overset(to)(b).overset(to)(c)=overset(to)(c).overset(to)(a)`
D. `overset(to)(a)xxoverset(to)(b)+overset(to)(b)xxoverset(to)(c)+overset(to)(c)xxoverset(to)(a)=overset(to)(0)`

Answer 1

Correct Answer - B
By triangle law `vec(a) + vec(b) + vec(c ) =vec(0)`

Taking cross product by `vec(a) , vec( b),vec( c)` respectively
`vec(a) xx (vec(a) + vec(b) + vec( c) ) = vec(a) xx vec(0) = vec(0)`
`rArr vec(a) xx vec(a) + vec(a) xx vec(b)+ vec(a) xx vec(c ) =vec(a)`
`rArr vec(a) xx vec(b) =vec(c ) xx vec(a)`
Similarly `vec(a) xx vec(b) = vec(b) xx vec(a)`
` :. vec(a) xx vec(b) = vec(b) xx vec(a)`