 Hello students, let's work out the following problem. It says if vector e is i cap plus 2 j cap minus 3 k cap and vector b is 3 i cap minus j cap plus 2 k cap Show that vector a plus vector b and vector a minus vector b are perpendicular to each other So let's now move on to the solution vector a is i cap plus 2 j cap minus 3 k cap and vector b is 3 i cap minus j cap plus 2 k cap Now we first find vector a plus vector b. Now 1 i cap plus 3 i cap is 4 i cap 2 j cap minus 1 j cap is plus 1 j cap minus 3 k cap plus 2 j cap is minus 1 k cap And this is the same as 4 i cap plus j cap minus k cap. Now we find vector a minus vector b Now i cap minus 3 i cap is minus 2 i cap 2 j cap Since we are subtracting sign will change. So this will become plus j cap So 2 j cap plus j cap is 3 j cap minus 3 k cap minus 2 k cap will be minus 5 k cap right Now we have to prove that vector a plus b and vector a minus vector b are perpendicular to each other. For that we'll prove that angle between these two vectors is pi by 2. Right? And we know that the angle between any two vectors given by cos theta if we have two vectors say x which is given by x i x 1 i cap x 2 j cap x 3 k cap and vector y as y 1 i cap plus y 2 j cap plus y 3 k cap Then angle between these two is given by cos theta And it is given by x 1 y 1 plus x 2 y 2 plus x 3 y 3 upon under the root x 1 square plus x 2 square plus x 3 square into under the root y 1 square plus y 2 square plus y 3 square You must remember this formula now We find angle between a vector a plus vector b and vector a minus vector b So cos theta is given by x 1 y 1 that is 4 into minus 2 plus x 2 y 2 that is 1 into 3 minus 1 into minus 5 upon under the root x 1 square that is 4 square plus x 2 square that is 1 square plus x 3 square that is minus 1 square into under the root y 1 square that is minus 2 square plus y 2 square that is 3 square plus y 3 square that is minus 5 square So now we have minus 8 plus 3 plus 5 upon under the root 4 square is 16 plus 1 plus 1 is 18 and this becomes 4 plus 9 plus 25 that is 38 and here we have minus 8 so this becomes 0 upon root 18 into root 38 which is equal to 0 so we have cos theta is equal to 0 so this implies theta is equal to pi by 2 thus angle between vector a plus vector b and vector a minus vector b is pi by 2 so this implies vector a plus vector b and vector a minus vector b are perpendicular to each other. This is the question and the session. Bye for now. Take care. Have a good day.