 Hello and welcome to the session. I am Deepika here. Let's discuss the question which says If the coordinates of the points a, b, c, d, b 1, 2, 3, 4, 5, 7 minus 4, 3, minus 6 and 2, 9, 2 respectively then find the angle between the lines a, b and c, d Now we know that direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axis and direction cosines of a line joining two points p with coordinates x1, y1, z1 and q with coordinates x2, y2, z2 are given by x2 minus x1 over pq y2 minus y1 over pq z2 minus z1 over pq where pq is equal to under root of 2 minus x1 whole square plus y2 minus y1 whole square plus z2 minus z1 whole square Again we know that the angle theta between two lines with direction cosines l1, m1, m1 and l2, m2, m2 is given by cos theta is equal to mod of l1, l2 plus m1, m2 plus m1, m2 Now these two formulas we will use in this question So this is the key idea behind our question So let's start the solution Now we are given the coordinates of the points a and b are 1, 2, 3 and 4, 5, 7 respectively So we are given a is a point with coordinates 1, 2, 3 and b is the point with coordinates 4, 5, 7 So according to our key idea a, b is equal to under root of 4 minus 1 whole square plus 5 minus 2 whole square plus 7 minus 3 whole square And this is equal to under root of 3 square which is 9 plus 3 square which is again 9 plus 4 square which is 16 And this is equal to under root of 34 Hence the direction cosines of the line joining a and v are 4 minus 1 over under root of 34, 5 minus 2 over under root of 34, 7 minus 3 over under root of 34 And this is equal to 3 over root 34, 3 over root 34, 4 over root 34 Again, again we are given c is the point with coordinates minus 4, 3 minus 6 And d is the point with coordinates 2, 9, 2 Therefore cd is equal to under root of 2 plus 4 whole square plus 9 minus 3 whole square Plus 2 plus 6 whole square which is equal to under root of 6 square which is 36 plus 6 square which is again 36 plus 8 square which is 64 And this is again equal to under root of 136 and this is equal to 2 root 34 Hence the direction cosines of the line joining c and d are 2 plus 4 over 2 root 34, 9 minus 3 over 2 root 34 2 plus 6 over 2 root 34 or 6 over 2 root 34 that is 3 over root 34 6 over 2 root 34 that is 3 over root 34, 8 over 2 root 34 that is 4 over root 34 Let theta is the acute angle between the lines ab and cd Therefore according to our key idea cos theta is equal to mod of 3 over root 34 into 3 over root 34 Which is 9 over 34 plus 3 over root 34 into 3 over root 34 which is again 9 over 34 Plus 4 over root 34 into 4 over root 34 which is 16 over 34 And this is equal to mod of 34 over 34 and this is again equal to 1 So cos theta is equal to 1 implies theta is equal to 0 degree because cos 0 is equal to 1 Hence the answer for the above question is 0 degree This means 2 lines ab and cd are parallel to each other So this completes our session, I hope the solution is clear to you Bye and have a nice day