 Hello and welcome to the session. Let's work out the following problem. It says the volume of the parallel of a bit Whose edges are minus 12 i cap plus lambda k cap 3 j cap minus k cap and 3 i cap plus j cap minus 15 k cap is 546 cubic units find the value of lambda. So let's now move on to the solution and let a be the Vector minus 12 i cap plus lambda k cap V be the vector 3 j cap minus k cap and C be the vector 2 i cap plus j cap minus 15 k cap. Now the volume of parallel of pippin is given by the formula mod of Vector A dot vector B cross vector C now Now we find vector A dot vector B cross vector C. Now this is the determinant The first row of this determinant is the coefficient of i cap j cap and k cap of the vector A So this is minus 12 the coefficient of j is 0 here lambda Then second row is the coefficient of i cap j cap and k cap of vector B Coefficient of i cap here is 0 then 3 minus 1 and the third row is the coefficient of i cap j cap and k cap of vector C this 2 1 minus 15 We are given that the volume of the parallel of pippin is 546 cubic units. Now I will expand this determinant So we have minus 12 into minus 15 into 3 minus of minus 1 minus 0 into 0 into 15 is 0 minus of minus 2 plus lambda into 0 into 1 is 0 minus 6 and this is equal to 546 So now we have minus 12 into minus 15 into 3 is minus 45 Plus 1 The second term would be 0 plus 0 plus lambda into minus 6 is minus 6 lambda is equal to 546 Now this implies minus 12 into minus 44 minus 6 lambda is equal to 546 minus 12 into minus 44 is 528 minus 6 lambda is equal to 546 this implies minus 6 lambda is Equal to 546 minus 528 minus 6 lambda is equal to 18. So this implies lambda is equal to 18 upon minus 6 and This implies lambda is equal to minus 3. So the value of lambda is minus 3 You must remember the formula for the volume of the parallel of paper. It is given by the mod of the vector a dot vector B cross vector C So this completes the question and the session by for now take care. Have a good day