The volume of a parallelopiped with coterminous edges a , b , c is 3 cubic units. The volume (in cubic units) of a tetrahedron with coterminous edges ( a b ), ( a 2 c ), ( b 2 c ) is...

Step-by-step solution

The volume of the parallelopiped with coterminous edges a , b , c is given by |[ a b c ]| = 3 . The volume of a tetrahedron with coterminous edges x , y , z is V = 1 6 |[ x y z ]| . Here, the edges are x = a b , y = 2( a c ) , and z = 2( b c ) . Evaluating the scalar triple product of these edges: [ x y z ] = [ a b 2( a c ) 2( b c )] = 4 [ a b a c b c ] Using the scalar triple product expansion: [ a b a c b c ] = ( a b ) (( a c ) ( b c )) Applying the vector triple product formula: ( a c ) ( b c ) = [ a c c ] b - [