 Hello and welcome to the session. In this session we discussed the following question which says if vector a is equal to i cap plus j cap plus k cap and vector b is equal to j cap minus k cap, find a vector c such that vector a cross vector c is equal to vector b and vector a road vector c is equal to 3. Now let's move on to the solution. We are given vector a equal to i cap plus j cap plus k cap vector b equal to j cap minus k cap and we need to find the vector c. So we take let vector c be equal to c 1 i cap plus c 2 j cap plus c 3 k cap. Now it is given to us that vector a cross vector c is equal to vector b. Now let's find out vector a cross vector c this is equal to determinant i cap j cap k cap 1 1 1 c 1 c 2 c 3. Now since we have vector a cross vector c so first we write the components of vector a and then we write the components of vector c and this is equal to vector b that is j cap minus k cap. Now this implies that i cap into c 3 minus c 2 minus j cap into c 3 minus c 1 plus k cap into c 2 minus c 1 is equal to 0 i cap plus j cap minus k cap. Then we have comparing coefficients of i cap j cap and k cap from both sides we get c 3 minus c 2 is equal to 0 then minus c 3 plus c 1 is equal to 1 then c 2 minus c 1 is equal to minus 1. Now from this equation we get c 3 is equal to c 2 let this be equation 1 then from this we get c 1 minus c 3 is equal to 1 let this be equation 2 and from this equation we get c 1 minus c 2 is equal to 1 and we take this as equation 3. The other condition given to us in the question is vector a or vector c is equal to 3 that is we have vector a dot vector c is equal to 3 this gives us i cap plus j cap plus k cap dot c 1 i cap plus c 2 j cap plus c 3 k cap is equal to 3. On taking the dot product of these two vectors we get c 1 plus c 2 plus c 3 is equal to 3. Now since we have from equation 1 c 3 is equal to c 2 so this can be written as c 1 plus c 2 plus c 2 is equal to 3 that is from 1 so this gives us c 1 plus 2 c 2 is equal to 3. We take this as equation 4 then we subtract equation 3 from equation 4 we get c 1 plus 2 c 2 minus c 1 minus c 2 is equal to 3 minus 1 that is this gives us c 1 plus 2 c 2 minus c 1 plus c 2 is equal to 2 that is we have 3 c 2 is equal to 2 which gives us c 2 is equal to 2 upon 3. Now we know that c 3 is equal to c 2 so we get that c 3 is also equal to 2 upon 3. Now we are just left to find the value for c 1 from equation 2 we have c 1 minus c 3 is equal to 1 on putting the value of c 3 equal to 2 upon 3 in this we get c 1 is equal to 1 plus 2 upon 3 and this is equal to 5 upon 3. So we get c 1 is equal to 5 upon 3 c 2 is equal to 2 upon 3 and c 3 is equal to 2 upon 3. Thus on substituting the values for c 1 c 2 c 3 vector c we get vector c is equal to c 1 i cap that is 5 upon 3 i cap plus c 2 j cap that is 2 upon 3 j cap plus c 3 k cap that is 2 upon 3 k cap. Hence vector c is equal to 1 upon 3 into 5 i cap plus 2 j cap plus 2 k cap. Hence our final answer is vector c is equal to 1 upon 3 into 5 i cap plus 2 j cap plus 2 k cap such that vector a cross vector c is equal to vector b and vector a dot vector c is equal to 3. So this completes the session hope you have understood the solution for this question.