 Good morning friends I am Purva and today we will work out the following question. If vector A is equal to 2i cap plus 2j cap plus 3k cap, vector B is equal to minus i cap plus 2j cap plus k cap and vector C is equal to 3i cap plus j cap are such that vector A plus lambda vector B is perpendicular to vector C then find the value of lambda. Let us now begin with the solution. Now we are given that vector A is equal to 2i cap plus 2j cap plus 3k cap, vector B is equal to minus i cap plus 2j cap plus k cap and vector C is equal to 3i cap plus j cap then we have vector A plus lambda vector B is equal to now vector A is equal to 2i cap plus 2j cap plus 3k cap plus lambda times vector B is equal to minus i cap plus 2j cap plus k cap this is equal to 2 minus lambda i cap plus 2 plus 2 lambda j cap plus 3 plus lambda k cap. So we have got vector A plus lambda vector B is equal to 2 minus lambda i cap plus 2 plus 2 lambda j cap plus 3 plus lambda k cap now we are given that vector A plus lambda vector B is perpendicular to vector C. So since vector A plus lambda vector B is perpendicular to vector C therefore we have the dot product of these two vectors is equal to 0. So we have vector A plus lambda vector B dot vector C is equal to 0. Now this implies vector A plus lambda vector B is equal to 2 minus lambda i cap plus 2 plus 2 lambda j cap plus 3 plus lambda k cap dot vector C is equal to 3i cap plus j cap and we have this is equal to 0. Now this further implies 3 into 2 minus lambda plus 1 into 2 plus 2 lambda plus 0 into 3 plus lambda is equal to 0 and this is since we have i cap dot i cap is equal to j cap dot j cap which is equal to k cap dot k cap is equal to 1 and i cap dot j cap is equal to j cap dot k cap is equal to k cap dot i cap is equal to 0 we know this. This implies now 3 into 2 gives 6 minus 3 into lambda is 3 lambda plus 1 into 2 is 2 plus 1 into 2 lambda is 2 lambda and this gives us 0 so we have this is equal to 0 and this further implies 6 plus 2 is 8 and minus 3 lambda plus 2 lambda gives minus lambda is equal to 0 this implies lambda is equal to 8 so we have got the value of lambda as 8 hence we write our answer as 8 hope you have understood the solution bye and take care.