 Hello and welcome to the session. In this session we discussed the following question that says find the shortest distance between the following lines x minus 3 upon 1 equal to y minus 5 upon minus 2 equal to z minus 7 upon 1 and x plus 1 over 7 equal to y plus 1 over minus 6 equal to z plus 1 over 1. Consider a line L1 x minus x1 over A1 is equal to y minus y1 over B1 is equal to z minus z1 over C1 and a line L2 x minus x2 over A2 is equal to y minus y2 over B2 is equal to z minus z2 over C2. Then we have the shortest distance between the lines L1 and L2 is given by determinant x2 minus x1 y2 minus y1 z2 minus z1 A1 B1 C1 A2 B2 C2 over square root of B1 C2 minus B2 C1 square plus C1 A2 minus C2 A1 square plus A1 B2 minus A2 B1 square. This is the key idea that we use for this question. Now let's move on to the solution. We are given the lines L1 and L2 where L1 is of the form x minus 3 over 1 equal to y minus pi over minus 2 equal to z minus 7 over 1 and L2 is of the form x plus 1 over 7 equal to y plus 1 over minus 6 equal to z plus 1 over 1. We need to find the shortest distance between L1 and L2. Let's name this equation of line as 1 and this as 2. Now comparing the given equations 1 and 2 with x minus x1 over A1 equal to y minus y1 over B1 equal to z minus z1 over C1 and x minus x2 over A2 equal to y minus y2 over B2 equal to z minus z2 over C2. We get that x1 is equal to 3, y1 is equal to 5, z1 is equal to 7, A1 is equal to 1, B1 is equal to minus 2, C1 is equal to 1 then we also have x2 is equal to minus 1, y2 is equal to minus 1, z2 is equal to minus 1 then A2 is equal to 7, B2 is equal to minus 6, C2 is equal to 1. Now we know that the shortest distance between the lines L1 and L2 is given by determinant x2 minus x1, y2 minus y1, z2 minus z1, A1 B1 C1, A2 B2 C2 whole upon square root of B1 C2 minus B2 C1 whole square plus A2 C1 minus A1 C2 whole square plus A1 B2 minus A2 B1 whole square. Now we substitute the values for x1, x2, y1, y2, z1, z2, A1 B1 C1, A2 B2 C2 in this and we get the shortest distance between the lines L1 and L2. So this would be equal to determinant minus 1, minus 3, minus 1, minus 5, minus 1, minus 7, 1, minus 2, 1, 7, minus 6, 1 whole upon square root of minus 2 into 1 minus minus 6 into 1 whole square plus 7 into 1 minus 1 into 1 whole square plus 1 into minus 6 minus 7 into minus 2 whole square. This further is equal to determinant minus 4, minus 6, minus 8, 1, minus 2, 1, 7, minus 6, 1 whole upon square root of minus 2 plus 6 the whole square plus 7 minus 1 the whole square plus minus 6 plus 14 the whole square that is we get this is further equal to determinant minus 4, minus 6, minus 8, 1, minus 2, 1, 7, minus 6, 1 whole upon square root 4 whole square that is 16 plus 6 whole square that is 36 plus 8 whole square that is 64 this is further equal to determinant minus 4, minus 6, minus 8, 1, minus 2, 1, 7, minus 6, 1 whole upon square root 116. Now let's find out the value for this determinant this would be equal to minus 4 into minus 2 minus minus 6 minus of minus 6 into 1 minus 7 plus minus 8 into minus 6 minus of minus 14 so this would be further equal to minus 4 into minus 2 plus 6 plus 6 into minus 6 minus 8 into 8 this is equal to minus 16 minus 36 minus 64 and this is equal to minus 116 so minus 116 is the value of the given determinant thus this would be equal to minus 116 upon square root of 116 which is further equal to minus square root of 116 now we know that the distance cannot be negative so we take modulus of this value which would be equal to square root of 116 so square root of 116 is the shortest distance between the two given lines thus the final answer is the shortest distance between the given two lines is square root of 116 so this completes the session hope you have understood the solution for this question