 Hello and welcome to the session. In this session we discussed the following question which says express the vector vector a equal to 8i cap minus 7j cap minus 4k cap as sum of two vectors vector v and vector c such that one is parallel to the vector z equal to i cap plus j cap plus k cap and the other is perpendicular to this vector. Consider two vectors so vector a and vector b. Vector a is perpendicular to vector b if and only if vector a dot vector b is equal to 0 and vector a is parallel to vector b implies that vector a is equal to lambda into vector b where this lambda is some scalar. This is the key idea that we use for this question. Now we move on to the solution. We are given a vector a is equal to 8i cap minus 7j cap minus 4k cap and we need to express this vector a as the sum of two vectors b and c that is vector a is equal to vector b plus vector c such that one vector is parallel to the vector z. So we can say let vector b be parallel to the vector z where we have vector z is equal to i cap plus j cap plus k cap and the other vector is perpendicular to vector z. The other vector is vector c so this vector c is perpendicular to the vector z where again vector z is equal to i cap plus j cap plus k cap. Now as we have that vector b is parallel to the vector z so this implies that vector b is equal to lambda into vector z for some scalar lambda. Consider this equation vector a is equal to vector b plus vector c as equation one. So from equation one we have vector a is equal to in place of vector b we put lambda into vector z plus vector c. So this gives us vector c is equal to vector a minus lambda into vector z. The next condition is that vector c is perpendicular to vector z so this implies that vector c dot vector z is equal to 0. Now we have vector c as vector a minus lambda into vector z dot vector z this is equal to 0. So this further implies vector a dot vector z minus lambda into vector z dot vector z is equal to 0. Now we substitute for vector a and vector z so we get 8 i cap minus 7 j cap minus 4 k cap dot i cap plus j cap plus k cap minus lambda into i cap plus j cap plus k cap dot i cap plus j cap plus k cap is equal to 0. So further we have 8 minus 7 minus 4 minus lambda into 1 plus 1 plus 1 is equal to 0 that is we have so here we have minus 3 minus 3 lambda is equal to 0 this gives us 3 lambda is equal to minus 3 which gives us lambda is equal to minus 1 so we get the value for lambda. Now we have vector b is equal to lambda into vector z our value for lambda is minus 1 so minus 1 into vector z which is i cap plus j cap plus k cap so this means that we have vector b is equal to minus i cap minus j cap minus k cap and vector c is equal to vector a minus lambda into vector z vector a is 8 i cap minus 7 j cap minus 4 k cap minus of minus 1 that is we put the value for lambda as minus 1 into vector z which is i cap plus j cap plus k cap this gives us vector c is equal to 8 i cap minus 7 j cap minus 4 k cap plus i cap plus j cap plus k cap so from here we have vector c is equal to 9 i cap minus 6 j cap minus 3 k cap so this is the vector c so we have decomposed the vector a as the sum of vector b and vector c where we have vector a is equal to 8 i cap minus 7 j cap minus 4 k cap vector b is equal to minus i cap minus j cap minus k cap vector c is equal to 9 i cap minus 6 j cap minus 3 k cap and also we have a vector z which is equal to i cap plus j cap plus k cap and this vector b is parallel to vector z and vector c is perpendicular to vector z so this comes easy session hope you have understood the solution of this question