 Good morning friends and poor one today. We will discuss the following question show that the lines x minus 5 upon 7 is equal to y plus 2 upon minus 5 is equal to z upon 1 and x upon 1 is equal to y upon 2 is equal to z upon 3 are perpendicular to each other Let us begin with the solution now Now we are given line 1 as x minus 5 upon 7 is equal to y plus 2 upon minus 5 is equal to z upon 1 and line 2 is x upon 1 is equal to y upon 2 is equal to z upon 3 Now the direction ratios for the line 1 that is a1 b1 and c1 are 7 minus 5 and 1 and The direction ratios for the line 2 that is a2 v2 and c2 are 1 2 and 3 Now the angle between the two lines is given by cos theta is equal to mod of a1 a2 plus b1 b2 plus c1 c2 upon under root of a1 square plus b1 square plus c1 square into under root of a2 square plus b2 square plus c2 square Putting the values we get cos theta is equal to mod of 7 into 1 plus minus 5 into 2 plus 1 into 3 upon under root of 7 square plus minus 5 whole square plus 1 square into under root of 1 square plus 2 square plus 3 square Now for the lines to be at right angles cos theta should be equal to 0 Now for cos theta to be equal to 0 the numerator of this expression should be equal to 0 So we get this implies the numerator of the above expression should be 0 Thus we have The numerator of this expression is 7 into 1 plus minus 5 into 2 plus 1 into 3 and This is equal to 7 into 1 is 7 minus 5 into 2 is minus 10 and 1 into 3 is 3 and This is equal to 0 So we have got numerator to be equal to 0 now since numerator is 0 cos theta is equal to 0 and this implies theta is equal to 90 degree that is The lines are perpendicular This is our answer. Hope you have understood the solution. Bye and take care