 Hello and welcome to this session. In this session we are going to discuss properties of scalar triple product. The first property is the scalar triple product is independent of the positions of dot and cross. That is vector A cross vector B dot vector C is equal to vector A dot vector B cross vector C. That is the positions of dot and cross can be interchanged provided the cyclic order of the vectors remains unchanged. Let vector A vector B vector C form a vector triad in the right handed system representing in magnitude and direction the coterminous edges of the parallel pipe. Then its volume is given by vector B is equal to vector A cross vector B dot vector C also. Vector B vector C and vector A form a vector triad in the right handed system representing in magnitude and direction the coterminous edges of the same parallel pipe. And therefore volume that is vector V is equal to vector B cross vector C dot vector A which can also be written as vector A dot vector B cross vector C. Now we have got vector V is equal to vector A cross vector B dot vector C. Also vector V is equal to vector A dot vector B cross vector C. Therefore we have vector A cross vector B dot vector C is equal to vector A dot vector B cross vector C. That is the scalar triple product is independent of position of dot and cross. The second property is the scalar triple product of three vectors unaltered so long as the cyclic order of the vectors remain unchanged. That is vector A cross vector B dot vector C is equal to vector B cross vector C dot vector A is equal to vector C cross vector A dot vector B. In other words we can say that if vector A vector B and vector C are slightly commuted the value of the scalar triple product remains same. Let us assume vector A vector B vector C for our right handed system representing the coterminous edges of a rectangular parallel pipe of volume V. Then vector V is equal to vector A cross vector B dot vector C also vector B vector C vector A and vector C vector A vector B form a right handed system representing the coterminous edges. For rectangular parallel pipe of volume V then vector V is equal to vector B cross vector C dot vector A and vector V is also equal to vector C cross vector A dot vector B. So we have got vector V is equal to vector A cross vector B dot vector C also vector V is equal to vector B cross vector C dot vector A and vector V is also equal to vector C cross vector A dot vector B. Hence vector A cross vector B dot vector C is equal to vector B cross vector C dot vector A is equal to vector C cross vector A dot vector B. Or we can also write it as scalar triple product of vectors A B C is equal to scalar triple product of vectors B C A is equal to scalar triple product of vectors C A B. Now the third property is the scalar triple product changes in sign but not in magnitude when the cyclic order is changed. In other words we can say that the change of cyclic order of vectors in scalar triple product changes in sign of the scalar triple product but not in magnitude. That is scalar triple product of vectors A B C is equal to minus of scalar triple product of vectors B A C is equal to minus of scalar triple product of vectors C B A is equal to minus of scalar triple product of vectors A C B. We know that scalar triple product of vectors A, B, C is equal to vector A cross vector B dot vector C. Since we know that vector A cross vector B is equal to minus of vector B cross vector A, so we can write scalar triple product of vectors A, B, C is equal to minus of vector B cross vector A dot vector C. So we have scalar triple product of vectors A, B, C is equal to minus of scalar triple product of vectors B, A, C, month of equation as 1. Also, by the second property we know that scalar triple product of vectors A, B, C is equal to scalar triple product of vectors B, C, A is equal to scalar triple product of vectors C, A, B and we have scalar triple product of vectors B, C, A is equal to vector B cross vector C dot vector A. Else, we can write it as scalar triple product of vectors B, C, A is equal to vector B cross vector C can be written as minus of vector C cross vector B dot vector A. So we have scalar triple product of vectors B, C, A is equal to minus of scalar triple product of vectors C, B, A and scalar triple product of vectors C, A, B is equal to vector C cross vector A dot vector B can be written as scalar triple product of vectors C, A, B is equal to vector C cross vector A can be written as minus of vector A cross vector C dot vector B which is equal to scalar triple product of vectors C, A, B is equal to minus of scalar triple product of vectors A, C, B. Now, the equation scalar triple product of vector B, C, A is equal to minus of scalar triple product of vectors C, B, A as equation 2 and scalar triple product of vector C, A, B is equal to minus of scalar triple product of vectors A, C, B as equation 3. Now, some equations 1, 2 and 3 and from the result of property 2 we get scalar triple product of vectors A, B, C is equal to minus of scalar triple product of vectors B, A, is equal to minus of scalar triple product of vectors C, B, A is equal to minus of scalar triple product of vectors A, C, B. The first property is the scalar triple product vanishes if any two of its vectors are equal that is scalar triple product of vectors A, A, B is equal to 0, scalar triple product of vectors A, B, A is equal to 0 and scalar triple product of vectors B, A, A is also equal to 0. We know that the scalar triple product of vectors A, A, B is equal to vector A cross vector A dot vector B and we know that vector A cross vector A is equal to 0. So, we have 0 into vector B that is equal to 0. From second property we know that scalar triple product of vectors A, B, A is equal to scalar triple product of vectors A, A, B which is equal to 0. Therefore, we have scalar triple product of vectors A, B, A is equal to 0, scalar triple product of vectors B, A, A is equal to vector B dot vector A cross vector A which is equal to vector B into vector A cross vector A is 0. Therefore, vector B into 0 is equal to 0. Now the fifth property is for any three vectors A, B, C and scalar lambda we have scalar triple product of vectors lambda A, B, C is equal to lambda A into scalar triple product of vectors A, B, C as we have scalar triple product of vectors lambda A, B, C which can be written as lambda into vector A cross vector B dot vector C which can be written as lambda into vector A cross vector B dot vector C which is equal to lambda into vector A cross vector B dot vector C that is lambda into scalar triple product of vectors A, B, C. Therefore, scalar triple product of vectors lambda A, B, C is equal to lambda into scalar triple product of vectors A, B, C. The sixth property is the scalar triple product vanishes if any two of its vectors are parallel or collinear let B, C be three vectors such that vector A is parallel to vector B then we have vector A is equal to lambda times vector B where lambda is some scalar. Therefore, scalar triple product of vectors A, B, C is equal to scalar triple product of vectors lambda B, B, C. Using property five this can be written as lambda into scalar triple product of B, B, C which is equal to lambda into scalar triple product of B, B, C is equal to zero using property four. So, we have lambda into zero that is equal to zero the seventh property is if A, B, C, B are four vectors then scalar triple product of vectors A plus B, C, B is equal to scalar triple product of vectors A, C, B plus scalar triple product of vectors B, C, B. Here we have scalar triple product of vectors A plus B, C, B which is equal to vector A plus vector B cross vector C dot vector B and this is equal to vector A cross vector C plus vector B cross vector B dot vector B using the distribution law by again applying the distribution law we get vector A cross vector C dot vector B plus vector B cross vector C dot vector B and this is equal to vector A cross vector C dot vector B can be written as scalar triple product of vector A, C, B plus vector B cross vector C dot vector B can be written as scalar triple product of vector B, C, B therefore scalar triple product of vectors A plus B, C, B is equal to scalar triple product of vectors A, C, B plus scalar triple product of vectors B, C, B. Let us now discuss product of four vectors because we have scalar product of four vectors is given by vector A cross vector B vector B equal to the determinant containing elements vector A dot vector C vector B dot vector C vector A dot vector D vector B dot vector D. We have vector product of four vectors is vector C cross vector D equal to scalar triple product of vectors ABD minus of scalar triple product of vectors two vector D is also equal to scalar triple product of vectors ACD two vector B minus of scalar triple product of vectors BCD into vector A and then we have four points vector A vector B vector C and vector D are containers to the product of vectors BCD vector vectors CAD plus scalar triple product of vectors ABD equal to scalar triple product of vectors ABC this completes our session. Hope you enjoyed this session.