Find the volume of the parallelepiped whose adjacent edges are: - 3 i ^ + 7 j ^ + 5 k ^ , - 5 i ^ + 7 j ^ - 3 k ^ , 7 i ^ - 5 j ^ - 3 k ^ .

Let the given vectors be a → = - 3 i ^ + 7 j ^ + 5 k ^ , b → = - 5 i ^ + 7 j ^ - 3 k ^ , c → = 7 i ^ - 5 j ^ - 3 k ^ We know that, the volume of the parallelepiped whose adjacent edges are a → , b → , c → is given by [ a → b → c → ] = a → · ( b → × c → ) Now, b → × c → = i ^ j ^ k ^ - 5 7 - 3 7 - 5 - 3 = i ^ ( - 21 - 15 ) - j ^ ( 15 + 21 ) + k ^ ( 25 - 49 ) = - 36 i ^ - 36 j ^ - 24 k ^ ∴ [ a → b → c → ] = - 3 i ^ + 7 j ^ + 5 k ^ · - 36 i ^ - 36 j ^ - 24 k ^ = 108 - 252 - 120 = 264 Hence, the required volume of the parallelepiped is 264 cubic units .

If a → = - 2 i ^ - 2 j ^ + 4 k ^ , b → = - 2 i ^ + 4 j ^ - 2 k ^ and c → = 4 i ^ - 2 j ^ - 2 k ^ , then calculate a → · ( b → × c → ) .

Given vectors: a → = - 2 i ^ - 2 j ^ + 4 k ^ , b → = - 2 i ^ + 4 j ^ - 2 k ^ and c → = 4 i ^ - 2 j ^ - 2 k ^ We know that, from scalar triple product, we have: [ a → b → c → ] = a → · ( b → × c → ) Now, b → × c → = i ^ j ^ k ^ - 2 4 - 2 4 - 2 - 2 = i ^ ( - 8 - 4 ) - j ^ ( 4 + 8 ) + k ^ ( 4 - 16 ) = - 12 i ^ - 12 j ^ - 12 k ^ ∴ [ a → b → c → ] = - 2 i ^ - 2 j ^ + 4 k ^ · - 12 i ^ - 12 j ^ - 12 k ^ = 24 + 24 - 48 = 0 Hence, the given vectors are coplanar in other words, the vector a → is lies in the plane of the vectors b → and c → .