 Good morning friends. I am Purva and today we will work out the following question. Find the angle between the following pair of lines. x-2 upon 2 is equal to y-1 upon 5 is equal to z-3 upon minus 3 and x-2 upon minus 1 is equal to y-4 upon 8 is equal to z-5 upon 4. Now the angle between two lines in Cartesian form is given by cos theta is equal to mod of a-1 a-2 plus b-1 b-2 plus c-1 c-2 upon under root of a-1 square plus b-1 square plus c-1 square into under root of a-2 square plus b-2 square plus c-2 square. And here a's b's and c's are the direction ratios of the given lines. So this is the key idea behind our question. Let us begin with the solution now. Now we are given the two lines as x-2 upon 2 is equal to y-1 upon 5 which is equal to z-3 upon minus 3 and line 2 is x-2 upon minus 1 is equal to y-4 upon 8 which is equal to z-5 upon 4. Now the direction ratios for the line 1 that is a-1 b-1 and c-1 are 2, 5 and minus 3 and the direction ratios for line 2 that is a-2 b-2 and c-2 are minus 1, 8 and 4. Now by key idea we know that the angle between two lines is given by cos theta is equal to mod of a-1 a-2 that is 2 into minus 1 plus b-1 b-2 that is 5 into 8 plus c-1 c-2 that is minus 3 into 4 upon under root of a-1 square that is 2 square plus b-1 square that is 5 square plus c-1 square that is minus 3 square into under root of a-2 square that is minus 1 square plus b-2 square that is 8 square plus c-2 square that is 4 square and this is equal to mod of 2 into minus 1 is minus 2 plus 5 into 8 is 40 minus 3 into 4 is minus 12 upon under root of 2 square is 4 plus 5 square is equal to 25 and minus 3 square is equal to plus 9 into under root of minus 1 square is equal to 1 plus 8 square is equal to 64 plus 4 square is equal to 16 this is equal to mod of minus 2 plus 40 minus 12 gives 26 upon under root of 38 into under root of a-1 and this is further equal to 26 upon under root of 38 into 9 so we have got cos theta is equal to 26 upon 9 under root 38 and this implies theta is equal to cos inverse 26 upon 9 under root 38 this is our answer hope you have understood the solution bye and take care