 Hello and welcome to the session. Let's work out the following problem. It says if vectors a to i cap minus j cap plus k cap vector b i cap plus 2 j cap plus 3 k cap vector c 3 i cap plus lambda j cap plus 5 k cap our coplanar find the value of lambda. So let's now move on to the solution Vector a is to i cap minus j cap plus k cap Vector b is i cap plus 2 j cap plus 3 k cap vector c is 3 i cap plus lambda j cap plus 5 k cap Now we have to find the value of lambda if these three vectors are coplanar. Now if vector a b and c are coplanar then Their scalar triple product is zero that is a dot v cross c is zero Now a dot v cross c is given by the determinant first row is a coefficient of i j and k cap of the vector a so it is 2 minus 1 1 Second row is of the coefficient of i cap j cap and 3 k cap of vector b. So it is 1 2 3 and third row is 3 lambda 5 and this is equal to 0 Now we will expand this determinant. So we have 2 into 5 into 2 10 minus 3 lambda minus of minus 1 into 5 into 1 minus 3 into 3 that is 9 plus 1 into 1 into lambda is lambda minus 3 into 2 is 6 this is equal to 0. So we have 2 into 10 minus 3 lambda minus minus plus 5 minus 9 is minus 4 plus lambda minus 6 is equal to 0. Now this implies 2 into 10 is 20 minus 6 lambda minus 4 plus lambda minus 6 is equal to 0. Now 20 minus 4 minus 6 is 10 minus 6 lambda plus lambda is minus 5 lambda is equal to 0. So this implies 10 is equal to 5 lambda and this implies lambda is equal to 10 by 5 and this implies lambda is equal to 2 Hence the value of lambda is 2. So this completes the question and the session. Bye for now take care. Have a good day