 Hello, I am welcome to the session, I am Deepika here. Let's discuss the question which says find the angle between the lines whose direction ratios are a, b, c and b minus c, c minus a, a minus b. Now we know that the angle between the two lines is given by cos theta is equal to mod of a1, a2 plus b1, b2 plus c1, c2 over under root of a1 square plus b1 square plus c1 square into a2 square plus b2 square plus c2 square where a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines, theta is the acute angle between these two lines. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now we are given direction ratios of two lines which are a, b, c and b minus c, c minus a, a minus b. That is we are given the direction ratios of the first line r and the direction ratios of the second line r, b minus c, c minus a, a minus b. If theta is the angle between them then according to our key idea cos theta is equal to mod of a into b minus c plus b into c minus a plus c into a minus b over under root of a square plus b square plus c square into under root of b minus a whole square plus c minus a whole square plus a minus b whole square. Now let us first solve the numerator. Numerator here is a into b minus c plus b into c minus a plus c into a minus b and this is equal to a b minus a c plus b c minus a b plus c a minus b c or this is equal to plus cos theta is equal to 0. This implies theta is equal to 90 degree. Hence the angle between the lines whose direction ratios are a, b, c and b minus c, c minus a, a minus b is 90 degree. So this is the answer for that question. So this completes our session. I hope the solution is clear to you. Bye and have a nice day.