 Hi friends, I am Purva and today we will work out the following question. Find the direction cosines of the sides of the triangle whose vertices are 3,5,-4,-1,1,2 and-5,-5,-2. 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. The direction cosines of line passing through two points P whose coordinates are x1, y1, z1 and q whose coordinates are x2, y2, z2 are x2-x1 upon PQ, y2-y1 upon PQ, z2-z1 upon PQ. Where we have PQ is equal to under root of x2-x1 whole square plus y2-y1 whole square plus z2-z1 whole square. So, this is the key idea behind our question. Let us begin with the solution now. Now we are given the vertices of the triangle are 3,5,-4,-1,1,2 and-5,-5,-2. So, let AB 3,5,-4, BB-1,1,2 and CB-5,-5,-2. Then we have AB is equal to under root of. Now by key idea we know that if P and q have coordinates x1, y1, z1 and x2, y2, z2 respectively, then PQ is equal to under root of x2-x1 whole square plus y2-y1 whole square plus z2-z1 whole square. So we get AB is equal to under root of minus 1 minus 3 whole square plus 1 minus 5 whole square plus 2 plus 4 whole square. This is equal to under root of 16 plus 16 plus 36 which is equal to 2 under root 17. Thus the direction cosines of side AB of the triangle are minus 1 minus 3 upon AB that is minus 1 minus 3 upon 2 under root 17 comma 1 minus 5 upon AB that is 1 minus 5 upon 2 under root 17 comma 2 plus 4 upon AB that is 2 plus 4 upon 2 under root 17. That is we have minus 2 upon under root 17 comma minus 2 upon under root 17 comma 3 upon under root 17. Similarly we have BC is equal to under root of minus 5 plus 1 whole square plus minus 5 minus 1 whole square plus minus 2 minus 2 whole square. And this is equal to under root of 16 plus 36 plus 16 which is equal to 2 under root 17. Thus the direction cosines of side BC of the triangle are minus 5 plus 1 upon 2 under root 17 comma minus 5 minus 1 upon 2 under root 17 comma minus 2 minus 2 upon 2 under root 17. That is we have minus 2 upon under root 17 comma minus 3 upon under root 17 comma minus 2 upon under root 17. Also we have CA is equal to under root of 3 plus 5 whole square plus 5 plus 5 whole square plus minus 4 plus 2 whole square. And this is equal to under root of 64 plus 100 plus 4 which is equal to 2 under root 42. Thus the direction cosines of side CA of the triangle are 3 plus 5 upon 2 under root 42 comma 5 plus 5 upon 2 under root 42 comma minus 4 plus 2 upon 2 under root 42. That is 4 upon under root 42 comma 5 upon under root 42 comma minus 1 upon under root 42. Thus the direction cosines of the sides of the triangle are minus 2 upon under root 17 comma minus 2 upon under root 17 comma 3 upon under root 17 minus 2 upon under root 17 comma minus 3 upon under root 17 comma minus 2 upon under root 17 and 4 upon under root 42. comma 5 upon under root 42 comma minus 1 upon under root 42 this is our answer hope you have understood the solution bye and take care.