Consider points $$A,\,B,\,C$$   and $$D$$ with position vectors $$7\hat i - 4\hat j + 7\hat k,\,\hat i - 6\hat j + 10\hat k,\, - \hat i - 3\hat j + 4\hat k$$         and $$5\hat i - \hat j + 5\hat k$$   respectively. Then $$ABCD$$   is a :

Solution :
$$A = \left( {7,\, - 4,\,7} \right),\,B = \left( {1,\, - 6,\,10} \right),\,C = \left( { - 1,\, - 3,\,4} \right)$$          and $$D = \left( {5,\, - 1,\,5} \right)$$
$$\eqalign{ & AB = \sqrt {{{\left( {7 - 1} \right)}^2} + {{\left( { - 4 + 6} \right)}^2} + {{\left( {7 - 10} \right)}^2}} \cr & = \sqrt {36 + 4 + 9} \cr & = 7 \cr} $$
Similarly $$BC = 7,\,CD = \sqrt {41} ,\,DA = \sqrt {17} $$
$$\therefore $$ None of the options is satisfied.