 Hello and welcome to this session. In this session, first we are going to discuss scalar triple product. The products formed with three vectors are called triple products B, C are only three vectors. Then by inserting dot and B cross between vector A, vector B and vector C in the same alphabetical order, we will have the following types of product. First type of product we can have is vector A dot vector B dot vector C which is meaningless as vector A dot vector B is a scalar quantity and we cannot define dot product between a scalar and a vector quantity. Second we have vector A dot vector B cross vector C which is again meaningless. Since vector A dot vector B is a scalar quantity and we cannot define the cross product between a scalar and a vector quantity, next we have vector A cross vector B dot vector C which is meaningless. Since vector A cross vector B is a vector quantity and its dot product with vector C is a scalar quantity and this product is known as scalar triple product and similarly vector A cross vector B cross vector C is meaningless. And this product is known as vector triple product. Now let us define scalar triple product. Let A be C be three vectors. Then the scalar or dot product of one of the vectors and the cross product of the remaining two is called a scalar triple product. In other words we can say that for any three given vectors vector A cross vector B dot vector C is called the scalar triple product and is denoted by scalar triple product of vector A B C. Can also be written as and we know that scalar triple product of vector A B C is equal to vector A cross vector B dot vector C. Now we are going to discuss geometrical interpretation of scalar triple product. Genetrically the scalar triple product that is vector A cross vector B dot vector C represents the volume of the parallel pipe formed by the three vectors B C. As its coterminous edges, let us draw a parallel pipe whose coterminous edges O P O Q and O R are represented in magnitude and direction by vector A vector B and vector C respectively. And vectors A B C are three non-coplanar vectors. Now vector A cross vector B is a vector perpendicular to the plane of vector A and vector B. Here theta is the angle between vector A and vector B. Let theta that be the angle between vector C and vector A cross vector B then vector A cross vector B dot vector C is equal to modulus of vector A into modulus of vector B into sin of angle theta into n pi dot vector C. As we know that vector A cross vector B is equal to modulus of vector A into modulus of vector B into sin of angle theta into n cap. Now this can be written as modulus of vector A into modulus of vector B into sin of angle theta into n cap dot vector C which is equal to modulus of vector A into modulus of vector B into sin of angle theta into. Now n cap dot vector C can be written as modulus of unit vector n cap into modulus of vector C into curve of theta dash. Therefore scalar triple product of vector A B C which is represented by is equal to A into B into sin of theta into modulus of n cap is equal to 1. Now modulus of vector C is equal to C into curve of angle theta dash. Therefore scalar triple product of vector A B C is equal to area of parallelogram O P S Q that is into B into sin of angle theta into C into curve of theta dash that is modulus of integral sin C n O P S Q which can also be written as scalar triple product of vector A B C. Therefore scalar triple product of vector A B C is equal to area of base into height of the parallelopiped. Therefore scalar triple product of vector A B C is equal to volume of the parallelopiped formed by the vectors A B C. Now with score modulus edges the scalar triple product of vectors A B C which is equal to vector A cross vector B dot vector C or can also be written as vector A dot vector B cross vector C is also called the box product. Vector A cross vector B dot vector C is positive or negative accordingly as vector A vector B vector C form a right or left handed system. Now we are going to discuss how to express the scalar product vector A cross vector B dot vector C in terms of components. Let vector A be equal to A1 i cap plus A2 j cap plus A3 k cap vector B be equal to B1 i cap plus B2 j cap plus B3 k cap. And vector C be equal to C1 i cap plus C2 j cap plus C3 k cap then vector B cross vector C can be written in the determinant form as the determinant containing elements i cap j cap k cap B1 B2 B3 C1 C2 C3. Which is equal to i cap into B2 into C3 minus of B3 into C2 minus of j cap into B1 into C3 minus of B3 into C1 plus j cap into B1 into C2 minus of B2 into C1. Now vector A dot vector B cross vector C the vector A is equal to A1 i cap plus A2 j cap plus A3 k cap and vector B cross vector C is equal to i cap into B2 C3 minus B3 C2 minus of j cap into B1 C3 minus B3 C1 plus j cap into B1 C2 minus B2 C1. Is equal to A1 i cap plus A2 j cap plus A3 k cap dot i cap into B2 C3 minus of B3 C2 minus of j cap into B1 C3 minus of B3 C1 plus j cap into B1 C2 minus of B3 C3 minus of B3 C1. On solving this we get vector A dot vector B cross vector C is equal to A1 into B2 into C3 minus of C2 into B3 minus of A2 into B1 into C3 minus of C1 into B3. Let A3 into B1 into C2 minus of C1 into B2 it can also be written as scalar triple vector vectors ABP is equal to the determinant containing elements A1 A2 A3 B1 B2 B3 C1 C2 C3. Now we shall discuss condition of coplanarity of three vectors. The necessary and sufficient condition by three non-zero and non-parallel vectors ABP the coplanar is scalar triple product of vectors ABP is equal to zero. First we shall prove the necessary condition let ABP B3 non-zero non-parallel and coplanar vectors and it is to be proved that scalar triple product of vectors ABP is equal to zero. In other words we can say that the volume of the parallel pipet comprise them as coterminous edges is zero as ABC actually non-zero coplanar vectors. Therefore vector B cross vector C is a vector perpendicular to the plane of vector B and vector C as vector A is coplanar with vector B and vector C. Vector A is also perpendicular to vector B cross vector C. Therefore vector A dot vector B cross vector C is equal to zero or we can say scalar triple product of vectors ABP is equal to zero. That is we have dot product of vector A with vector B cross vector C is zero as for theta is equal to pi by 2 we have vector A dot vector B is equal to modulus of vector A into modulus of vector B into cos of theta. Which is equal to modulus of vector A into modulus of vector B into cos of pi by 2. Since cos of pi by 2 is equal to zero therefore we have vector A dot vector B is equal to zero. Now we shall prove the sufficient condition that is let scalar triple product of vectors ABC be equal to zero and we have to prove that vector A, vector B and vector C are coplanar. As we have assumed that scalar triple product of vectors ABC is equal to zero which implies that vector A dot vector B cross vector C is equal to zero and mark this equation as 1. This shows that either vector A is perpendicular to vector B cross vector C or vector A is equal to zero or vector B cross vector C is equal to zero but vector A cannot be equal to zero as it is given that vector A is a non-zero vector. Now vector B cross vector C is equal to zero implies that vector B is equal to zero vector C is equal to zero or vector B and vector C are parallel vectors but it is also given that vector B cannot be equal to zero. So vector C cannot be equal to zero and vector B and vector C are non-parallel vectors therefore vector B cross vector C cannot be equal to zero therefore equation 1 implies that vector A is perpendicular to vector B cross vector C but vector B cross vector C is perpendicular to the plane of vector B and vector C therefore vector A should be in the plane of vector B and vector C therefore vector A, vector B and vector C are coplanar vectors. This completes our session. Hope you enjoyed this session.