If four points ` A (overset(-)a), B(overset(-)b), C (overset(-)c) & D(overset(-)d) ` are coplanar then show that `[ overset (-)a overset (-)b overset(-)d ]+ [ overset (-) b overset (-)c overset (-)d ]+ [ overset (-) c overset (-)a overset (-)d ]= [overset (-)a overset (-)b overset (-)c ]`

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)

H_(3)overset(4)C-overset(O)overset(||)overset(3" ")(C)-overset(2)CH_(2)-overset(O)overset(||)overset(1" ")(C)-OH

If vectors overset(to)(a) , overset(to)(b) , overset(to)( C) are coplanar then show that |{:(overset(to)(a),,overset(to)(b),,overset(to)(c )),(overset(to)(a)"."overset(to)(a),,overset(to)(a)"."overset(to)(b),,overset(to)(a)"."overset(to)(c )),(overset(to)(b)"."overset(to)(a),,overset(to)(b)"."overset(to)(b),,overset(to)(b)"." overset(to)(c )):}|

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

If overset(to)(a) , overset(to)(b) , overset(to)(c ) " and " overset(to)(d) are the unit vectors such that (overset(to)(a)xx overset(to)(b)). (overset(to)(c )xx overset(to)(d)) =1 " and " overset(to)(a), overset(to)(c ) = .(1)/(2) , then

If overset(to)(a) , overset(to)(b) , overset(to)(c ) are non-coplanar unit vectors such that overset(to)(a) xx (overset(to)(b) xx overset(to)(c )) = ((overset(to)(b) + overset(to)(c )))/(sqrt(2)) , then the angle between overset(to)(a) " and " overset(to)(b) is

If overset(to)(a) , overset(to)(b) " and " overset(to)( c) are unit coplanar vectors then the scalar triple product [2 overset(to)(a) - overset(to)(b) 2 overset(to)(b) - overset(to)(c ) 2 overset(to)(c ) - overset(to)(a)] is

The correct IUPAC name of the H-overset(O)overset(||)C-overset(O)overset(||)C-overset(O)overset(||)C-OH is

If overset(to)(a),overset(to)(b),overset(to)(c ),overset(to)(d) are four distinct vectors satisfying the conditions overset(to)(a)xxoverset(to)(b)=overset(to)(c )xx overset(to)(d) " and " overset(to)(a)xxoverset(to)(c ) = overset(to)(b)xx overset(to)(d) then prove that , overset(to)(a).overset(to)(b)+overset(to)(c ). overset(to)(d) ne overset(to)(a). overset(to)(c)+overset(to)(b).overset(to)(d) .

For any three vectors overset(to)(a), overset(to)(b) " and " overset(to)(C ) (overset(to)(a) - overset(to)(b)). {(overset(to)(b)-overset(to)(c))xx(overset(to)(c)-overset(to)(a))} = 2overset(to)(a).(overset(to)(b)xx overset(to)(c))