 Hi friends, I am poor one today. We will work out the following question find the angle between the following pair of lines X upon 2 is equal to y upon 2 is equal to z upon 1 and X minus 5 upon 4 is equal to y minus 2 upon 1 is equal to z minus 3 upon 8 Suppose the direction ratios for line 1 are a 1 b 1 and c 1 and the direction ratios for line 2 are a 2 b 2 and C 2 then the angle between these 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 So this is the key idea behind our question Let us begin with the solution now Now we are given line 1 as X upon 2 is equal to y upon 2 is equal to z upon 1 and line 2 is X minus 5 upon 4 is equal to y minus 2 upon 1 Which is equal to z minus 3 upon 8 Now the direction ratios for line 1 are Now the direction ratios for line 1 are a 1 b 1 and c 1 and they are 2 2 and 1 and the direction ratios for line 2 That is a 2 b 2 and c 2 are 4 1 and 8 Now by key idea the angle between the two lines is given by cos theta is equal to mod of a 1 Into a 2 that is 2 into 4 plus v 1 into b 2 that is 2 into 1 Plus c 1 into c 2 that is 1 into 8 upon under root of a 1 square that is 2 square Plus b 1 square that is 2 square Plus c 1 square that is 1 square Into under root of a 2 square that is 4 square Plus b 2 square that is 1 square Plus c 2 square that is 8 square and this is equal to mod of 2 into 4 is 8 plus 2 into 1 is 2 plus 1 into 8 is 8 upon under root of 2 square is 4 plus 2 square is 4 Plus 1 square is 1 into under root of 4 square is 16 Plus 1 square is 1 plus 8 square is 64 and this is equal to 8 plus 2 plus 8 is equal to 18 upon under root of 4 plus 4 plus 1 is 9 into under root of 16 plus 1 plus 64 is 81 and This is equal to 18 upon Now under root of 9 is equal to 3. So we get 3 into and under root of 81 is equal to 9 So we get 9 So we get cos theta is equal to now canceling the common factors We get 2 upon 3 on the right-hand side So this implies Theta is equal to cos inverse 2 upon 3 This is our answer. Hope you have understood the solution. Bye and take care