If `overset(to)(X) "." overset(to)(A) =0, overset(to)(X) "." overset(to)(B) =0, overset(to)(X) "." overset(to)(C ) =0` for some non-zero vector `overset(to)(X) " then " [overset(to)(A) overset(to)(B) overset(to)(C )]=0`

Let vec(A),vec(B),vec(C ) be vectors of length 3, 4, 5, respectively Let overset(to)(A) be perpendicular to overset(to)(B) +overset(to)(C ) , overset(to)(B) " to " overset(to)( C) + overset(to)(A) " and " overset(to)(C ) to overset(to)(A) +overset(to)(B) then the length of vector overset(to)(A) +overset(to)(B)+overset(to)(C ) is .......

Let vec(A),vec(B),vec(C ) be vectors of length 3, 4, 5, respectively Let overset(to)(A) be perpendicular to overset(to)(B) +overset(to)(C ) , overset(to)(B) " to " overset(to)( C) + overset(to)(A) " and " overset(to)(C ) to overset(to)(A) +overset(to)(B) then the length of vector overset(to)(A) +overset(to)(B)+overset(to)(C ) is .......

The scalar overset(to)(A) .[(overset(to)(B) + overset(to)( C)) xx (overset(to)(A) + overset(to)(B) + overset(to)( C))] equals

The scalar overset(to)(A) .[(overset(to)(B) xx overset(to)( C)) xx (overset(to)(A) + overset(to)(B) + overset(to)( C))] equals

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)

If overset(to)(A) , overset(to)(B) " and " overset(to)( c) are vectors such that |overset(to)(B) |=|overset(to)( C ) | . Prove that | (overset(to)(A) + overset(to)(B)) xx (overset(to)(A) + overset(to)(C )) | xx (overset(to)(B) xx overset(to)(C )) . (overset(to)(B) + overset(to)( C )) = overset(to)(0)

Let overset(to)(A),overset(to)(B)" and " overset(to)(C ) be unit vectors . If overset(to)(A).overset(to)(B) = overset(to)(A).overset(to)(C ) =0 and that the angle between overset(to)(B) " and " overset(to)(C )" is " pi//6. Then overset(to)(A) =+-2 (overset(to)(B)xxoverset(to)(C ))

Let overset(to)(A),overset(to)(B)" and " overset(to)(C ) be unit vectors . If overset(to)(A).overset(to)(B) = overset(to)(A).overset(to)(C ) =0 and that the angle between overset(to)(B) " and " overset(to)(C )" is " pi//6. Then overset(to)(A) =+-2 (overset(to)(B)xxoverset(to)(C ))

Let overset(to)(p) , overset(to)(q) , overset(to)(r ) be three mutually perpendicular vectors of the same magnitude. If a vectors overset(to)(X) satisfies the equation overset(to)(p) xx [(overset(to)(x) -overset(to)(q)) xx overset(to)(p)] + overset(to)(q) xx [(overset(to)(x)-overset(to)(r ))xx overset(to)(q)] + overset(to)(r ) xx [(overset(to)(x) - overset(to)(p)) xx overset(to)(r )]=overset(to)(0) " then " overset(to)(x) is given by

If |overset(to)(a) + overset(to)(b)| =| overset(to)(a) - overset(to)(b) | , prove that overset(to)(a) and overset(to)(b) are perpendicular