 Hi and welcome to the session. Let us discuss the following question, question says show that A vector minus B vector cross A vector plus B vector is equal to 2 multiplied by A vector cross B vector. Let us now start with the solution. Now we have to show that A vector minus B vector cross A vector plus B vector is equal to 2 multiplied by A vector cross B vector. First of all let us consider the vacant side of this expression. Vacant side is A vector minus B vector cross A vector plus B vector. Now to find this cross product we will use distributivity of vector product over addition. Now using this property we get this expression is equal to A vector minus B vector cross A vector plus A vector minus B vector cross B vector. Here we have replaced P by A vector minus B vector and Q vector has been replaced by A vector here and R vector has been replaced by B vector here. Now this is further equal to A vector plus minus B vector cross A vector plus A vector plus minus B vector cross B vector. Now again we will apply this property here and we get A vector cross A vector plus minus B vector cross A vector we will write this plus sign as it is. Now applying this property to this part we get A vector cross B vector plus minus B vector cross B vector. Now applying this property we get this expression is equal to A vector cross A vector minus B vector cross A vector plus A vector cross B vector minus B vector cross B vector we have used this property in these two cross products. Now using this property of vector product we get A vector cross A vector is equal to 0. Similarly here we get B vector cross B vector is equal to 0 and we will write minus B vector cross A vector plus A vector cross B vector as it is and we will replace these two cross products by 0. Now using this property of vector product we can write minus vector B cross vector A as A vector cross B vector and we will write plus A vector cross B vector as it is. Now clearly we can see these two vectors are same. Now using this property we get addition of these two vectors is equal to 2 multiplied by A vector cross B vector. Here we have replaced C vector by A vector cross B vector and value of k is equal to 1 and value of m is also equal to 1 here. Now we get 1 plus 1 is equal to 2 and we know C vector is equal to A vector cross B vector. So here we have written A vector cross B vector. Now 2 multiplied by A vector cross B vector is equal to right hand side of the given expression. So this is our required answer. Hence proved this completes the session. Hope you understood the solution. Take care and have a nice day.