 Hello and welcome to the session. In this session we discuss the following question which says if vector A cross vector B is equal to vector C cross vector D and vector A cross vector C is equal to vector B cross vector D show that vector A minus vector D is parallel to vector B minus vector C where vector A is not equal to vector D and vector B is not equal to vector C. Before we move on to the solution let's recall the condition when two given vectors are parallel to each other. So if we have vector A and vector B as two non-zero vectors then vector A cross vector B is equal to zero vector if and only if vector A is parallel to the vector B. This is the key idea that we use in this question. Let's move on to the solution now. We are given vector A cross vector B is equal to vector C cross vector D. Let this be equation one then we also have vector A cross vector C is equal to vector B cross vector D. Let this be equation two. Now next subtracting equation two from equation one we get vector A cross vector B minus vector A cross vector C is equal to vector C cross vector D minus vector B cross vector D. So this further gives us vector A cross vector B minus vector C the whole is equal to vector C minus vector B the whole cross vector D. So now we have vector A cross vector B minus vector C the whole minus vector C minus vector B the whole cross vector D is equal to zero vector. Since we know that vector A cross vector B is equal to minus vector B cross vector A so this would further give us vector A cross vector B minus vector C the whole plus vector D cross vector C minus vector B the whole is equal to zero vector or you can say vector A cross vector B minus vector C the whole minus vector D cross vector B minus vector C the whole is equal to zero vector. This gives us vector A minus vector D the whole cross vector B minus vector C is equal to zero vector. Now since the cross product of these two vectors is a zero vector and according to the condition given in the key idea for the two vectors to be parallel is that the cross product should be a zero vector. So this means that vector A minus vector D is parallel to vector B minus vector C as we are also given that vector A is not equal to vector D and vector B is also not equal to vector C. So we have proved that vector A minus vector D is parallel to vector B minus vector C so hence proved this completes the session hope you have understood the solution of this question.