 Hi friends, I am Purva and today we will work out the following question. If a line has the direction ratios minus 8 comma 12 comma minus 4, then what are its direction cosines? If a directed line L passing through the origin makes angles alpha, beta and gamma with x, y and z axis respectively, then cosines of these angles namely cos alpha, cos beta and cos gamma are called direction cosines of the directed line L. Now, let a comma b comma c be the direction ratios of the line. L comma m comma n be the direction cosines. Then we get L is equal to plus minus a upon under root of a square plus b square plus c square m is equal to plus minus b upon under root of a square plus b square plus c square and n is equal to plus minus c upon under root of a square plus b square plus c square. So, this is the key idea behind the question. Let us now begin with the solution. Now, we are given that direction ratios are minus 18 comma 12 comma minus 4. Now, by key idea, let a comma b comma c be the direction ratios, then we get thus a is equal to minus 18, b is equal to 12 and c is equal to minus 4. Now, we have to find the direction cosines. Now, by key idea, we know that L, m, n are the direction cosines and they are given by L is equal to plus minus a upon under root of a square plus b square plus c square and this is equal to now a is equal to minus 18. So, we have minus 18 upon under root of minus 18 square plus 12 square plus minus 4 square and this is further equal to minus 18 upon under root of now minus 18 square is 324 plus 12 square is equal to 144 plus minus 4 square is equal to 16. This is equal to minus 18 upon under root of 484 which is further equal to minus 18 upon 22 and cancelling common factor 2, we get this implies L is equal to minus 9 upon 11. Now, m is given by m is equal to plus minus b upon under root of a square plus b square plus c square and this is equal to 12 upon under root of minus 18 square plus 12 square plus minus 4 square. This is equal to 12 upon under root 484 which is equal to 12 upon 22. Now, cancelling common factor 2, we get this implies m is equal to 6 upon 11. Now finally, n is given by n is equal to plus minus c upon under root of a square plus b square plus c square and we get this is equal to minus 4 upon under root of minus 18 square plus 12 square plus minus 4 square and this is equal to minus 4 upon under root of 484 which is further equal to minus 4 upon 22. Now, cancelling the common factor 2, we get this implies n is equal to minus 2 upon 11. Thus, the direction cosines are minus 9 upon 11, 6 upon 11 and minus 2 upon 11. This is our answer. Hope you have understood the solution. Bye and take care.