 Hello and welcome to the session, let us discuss the following question. It says, prove that scalar triple product of vector a plus vector b, vector b plus vector c, vector c plus vector a is equal to twice of the scalar triple product of vector a, vector b and vector c and hence show that if a vector a, b, c are non-coplanar then vector a plus vector c, vector c plus vector a are also non-coplanar. So, let us now move on to the solution. We have to prove that the scalar triple product of vector a plus vector b, vector b plus vector c, vector c plus vector a is equal to this. So, let us start with this and this is equal to vector a plus vector b cross vector b plus vector c dot product vector c plus vector a. This is by the definition of scalar triple product that is if we have three vectors u, v and w then it is given by vector u cross vector v dot vector w. Now, we will use the distributed law. So, we have vector a cross vector v plus vector a cross vector c plus vector b cross vector v and vector v cross vector c dot vector c plus vector a and this is again equal to vector a cross vector b plus vector a cross vector c plus vector b cross vector c. As we know that the cross product of two equal vectors is 0. So, vector b cross vector b is 0 dot product vector c plus vector a. Here we have used distributed law and here vector b cross vector v is 0. Now, again we will use the distributed law. So, we have vector a cross vector b dot vector c plus vector a cross vector c dot vector c plus vector b cross vector c dot vector c plus vector a cross vector b dot vector a plus vector a cross vector c dot vector a plus vector b cross vector c dot vector a. Now, this is again can be written as scalar triple product of vector a vector b vector c and this is scalar triple product of vector a vector c vector c and this is scalar triple product of vector b vector c vector c and this is scalar triple product of vector a vector b vector a plus vector a, vector c, vector a and this is the scalar triple product of vector b, vector c, vector a. Now we know that the scalar triple product when any two vectors are equal is 0. So, this will be 0, this will be 0, this will be 0 and this will also be 0. So, we have scalar triple product of vector a, vector b, vector c plus scalar triple product of vector b, vector c, vector a. This is because scalar triple product when any two vectors are equal is 0. Now, scalar triple product of vector a, vector b, vector c plus vector scalar triple product of vector b, vector c, vector a is twice of the scalar triple product of vector a, vector b, vector c. This is because scalar triple product of vector a, vector b, vector c is same as scalar triple product of vector B vector C vector A. The scalar triple product of vector A plus vector B vector B plus vector C vector C plus vector A is equal to twice of scalar triple product of vector A vector B vector C. Now we have to show that if vector A vector B vector C are non-coplanar then vector A plus vector B vector B plus vector C vector C plus vector A are also non-coplanar. So we are given vector A vector B vector C are non-coplanar. Therefore, scalar triple product of vector A vector B vector C is non-zero. So this implies twice of scalar triple product of vector A vector B vector C is non-zero. But twice of scalar triple product of vector A vector B vector C is scalar triple product of vector A plus vector B vector B plus vector C vector C plus vector A and since this is non-zero this is also non-zero. So this implies vector A plus vector B vector B plus vector C vector C plus vector A are also non-coplanar hence proved. So this completes the question and this session by for now take care have a good day.