 Hi friends, I am Purva and today we will work out the following question. Show that the three lines with direction cosines 12 upon 13 comma minus 3 upon 13 comma minus 4 upon 13, 4 upon 13 comma 12 upon 13 comma 3 upon 13 and 3 upon 13 comma minus 4 upon 13 comma 12 upon 13 are mutually perpendicular. 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. Let the direction cosines of two lines L1 and L2 be L1, M1, N1 and L2, M2, N2. Then the angle theta between them is given by cos theta is equal to mod of L1, L2 plus M1, M2 plus N1, N2. These two lines are perpendicular if theta is equal to 90 degrees. So this is the key idea behind our question. Let us begin with the solution now. Now we are given the direction cosines for L1R, 12 upon 13 comma minus 3 upon 13 comma minus 4 upon 13 for L2R, 4 upon 13 comma 12 upon 13 comma 3 upon 13 and for L3R, 3 upon 13 comma minus 4 upon 13 comma 12 upon 13. Let angle between L1 and L2 be theta 1, then cos theta 1 is equal to mod of, now by key idea we know that cos theta is equal to mod of L1, L2 plus M1, M2 plus N1 into, so cos theta 1 is equal to mod of 12 upon 13 into 4 upon 13 plus minus 3 upon 13 into 12 upon 13 plus minus 4 upon 13 into 3 upon 13. This is equal to mod of 48 upon 169 minus 36 upon 169 minus 12 upon 169 and this is equal to 0. So we have got cos theta 1 is equal to 0. This implies theta 1 is equal to 90 degrees. Similarly, let angle between L2 and L3 be theta 2, then cos theta 2 is equal to mod of 4 upon 13 into 3 upon 13 plus 12 upon 13 into minus 4 upon 13 plus 3 upon 13 into 12 upon 13. This is equal to mod of 12 upon 169 minus 48 upon 169 plus 36 upon 169 and this is equal to 0. So we have got cos theta 2 is equal to 0. This implies theta 2 is equal to 90 degrees. Now let angle between L1 and L3 be theta 3, then cos theta 3 is equal to mod of 12 upon 13 into 3 upon 13 plus minus 3 upon 13 into minus 4 upon 13 plus minus 4 upon 13 into 12 upon 13. This is equal to mod of 36 upon 169 plus 12 upon 169 minus 48 upon 169 and this is equal to 0. So we have got cos theta 3 is equal to 0 which implies theta 3 is equal to 90 degrees. Now by key idea we know that two lines are perpendicular if theta is equal to 90 degrees. So since theta 1 is equal to theta 2 is equal to theta 3 is equal to 90 degrees. Therefore the three lines are perpendicular to each other. This is our answer. Hope you have understood the solution. Bye and take care.