 Hello and welcome to the session. In this session we are going to discuss Condition for Prependicularity and Paranism for two given lines. When the given lines are prependicular to each other then angle between them is 90 degrees that is theta is equal to 90 degrees which implies that the value of cos of theta that is equal to cos of 90 degrees is equal to 0 but cos of theta is given by L1 into L2 plus M1 into M2 plus M1 into M2. When the value of theta is equal to 90 degrees then we have cos of 90 degrees is equal to L1 into L2 plus M1 into M2 plus M1 into M2 which implies that cos of 90 degrees that is 0 is equal to L1 into M2 plus M1 into M2 plus M1 into M2 which implies that L1 into L2 plus M1 into M2 plus M1 into M2 is equal to 0 is the required condition of prependicularity of two given lines where L1 M1 M1 and L2 M2 M2 are the direction for the two given lines also if A1 B1 C1 and A2 B2 C2 are the direction ratios of the two lines then cos of theta is equal to plus minus of A1 into A2 plus B1 into B2 plus C1 into C2 upon square root of summation of A1 square into square root of summation of A2 square where summation of A1 square is equal to A1 square plus B1 square plus C1 square and summation of A2 square is equal to A2 square plus B2 square plus C2 square and if the value of theta is given as 90 degrees then cos of angle theta that is 90 degrees is equal to 0 therefore cos of 90 degrees which is equal to 0 is equal to plus minus A1 into A2 plus B1 into B2 plus C1 into C2 upon square root of summation of A1 square into square root of summation of A2 square which implies that plus minus of A1 into A2 plus B1 into B2 plus C1 into C2 is equal to 0 that is the value of A1 into A2 plus B1 into B2 plus C1 into C2 is equal to 0 which is also the condition of preventive polarity of two lines looking we have when the given lines are parallel to each other that is the value of theta is equal to 0 degrees so sin of theta that is sin of 0 degrees is equal to 0 which implies that sin square of angle theta is equal to 0 we know that the value of sin square of theta is equal to 1 minus cos square of theta and cos of theta is given by L1 into L2 plus L1 into L2 plus L1 into L2 which implies that sin square of theta is equal to 1 minus cos square of theta that is L1 into L2 plus L1 into L2 plus L1 into L2 B whole square on serving this we get sin square theta is equal to L1 into L2 minus of L2 into L1 B whole square plus L1 into L2 minus of L2 into L1 B whole square plus L1 into L2 minus of L2 into L1 B whole square sin square theta is equal to 0 implies that L1 into L2 minus of L2 into L1 B whole square plus L1 into L2 minus of L2 into L1 B whole square plus L1 into L2 minus of L2 into L1 B whole square is equal to 0 As sum of these squares of non-zero terms cannot be 0. Therefore, L1 into M2 minus of L2 into M1 is equal to 0. M1 into M2 minus of M2 into M1 is equal to 0. And L1 into L2 minus of M2 into L1 is equal to 0. L1 into M2 minus of L2 into M1 is equal to 0 implies that L1 into M2 is equal to L2 into M1. Similarly, M1 into M2 minus of M2 into M1 is equal to 0 implies that M1 into M2 is equal to M2 into M1 and M1 into L2 minus of M2 into L1 is equal to 0 implies that M1 into L2 is equal to M2 into L1. Which server implies that L1 into M2 is equal to L2 into M1 can be written as L1 by L2 is equal to M1 by M2. Similarly, M1 into M2 is equal to M2 into M1 can be written as M1 by M2 is equal to M1 by M2 and M1 into L2 is equal to M2 into L1 can be written as M1 by M2 is equal to L1 by L2. Now, L1 by L2 is equal to M1 by M2, M1 by M2 is equal to M1 by M2 and M1 by M2 is equal to L1 by L2 implies that L1 by L2 is equal to M1 by M2 is equal to M1 by M2 also is A1, B1, C1 and A2, B2, C2 are the direction ratio of the two lines as lines are parallel, theta is equal to 0 degrees then we have sin of theta that is sin of 0 degrees is equal to 0 which implies that sin square of theta will also be equal to 0. Now, as we know that the value of sin square of theta is given by L1 into M2 minus of L2 into M1 B whole square plus M1 into M2 minus of M2 into M1 B whole square plus M1 into L2 minus of M2 into L1 B whole square and we know that the value of L1 is given by A1 upon square root of summation of A1 square M1 is equal to B1 upon square root of summation of A1 square and M1 is equal to C1 upon square root of summation of A1 square Similarly, A2 is equal to A2 upon square root of summation of A2 square into is equal to B2 upon square root of summation of A2 square and n2 is equal to c2 upon square root of summation of a2 square. The summation of a1 square is equal to a1 square plus b1 square plus c1 square and summation of a2 square is equal to a2 square plus b2 square plus c2 square. Where l1, m1, n1 and l2, m2, n2 are the direction cosines of the two lines and a1, b1, c1 and a2, b2, c2 are the direction ratio of the two lines and I am putting the values of l1, m1, n1 and l2, m2, n2 in this equation we get a1 into b2 minus of a2 into b1 b2 square plus b1 into c2 minus of b2 into c1 b2 square plus c1 into a2 minus of c2 into a1 b2 square whole upon square root of summation of a1 square into square root of summation of a2 square is equal to 0 which implies that a1 into b2 minus of a2 into b1 b2 square plus b1 into c2 minus of b2 into c1 b2 square plus c1 into a2 minus of c2 into a1 b2 square is equal to 0 and this is true if each term is equal to 0 therefore we have a1 into b2 minus of a2 into b1 is equal to 0 b1 into c2 minus of b2 into c1 is equal to 0 and c1 into a2 minus of c2 into a1 is equal to 0 now a1 into b2 minus of a2 into b1 is equal to 0 implies that a1 by a2 is equal to b1 by b2 similarly b1 into c2 minus of b2 into c1 is equal to 0 implies that b1 by b2 is equal to c1 by c2 and c1 into a2 minus of c2 into a1 is equal to 0 implies that c1 by c2 is equal to a1 by a2 now we have a1 by a2 is equal to b1 by b2 b1 by b2 is equal to c1 by c2 and c1 by c2 is equal to a1 by a2 which implies that a1 by a2 is equal to b1 by b2 is equal to c1 by c2 which is also a condition for parallelism let us take an example so that the lines join in a with the coordinates 10 4 minus 6 to b with the coordinates 12 to 8 and c with the coordinates minus 6 minus 4 minus 2 to b with the coordinates minus 2 minus 8 26 are parallel here we have given point a with the coordinates 10 4 minus 6 and point b with the coordinates 12 to 8 then the direction ratios of line a b are given by 12 minus 10 that is 2 2 minus 4 that is minus 2 8 minus of minus 6 that is 14 so the value of a1 is equal to 2 the value of b1 is equal to minus 2 and the value of c1 is equal to 14 similarly we are given point c with the coordinates minus 6 minus 4 minus 2 and point b with the coordinates minus 2 minus 8 26 then the direction ratios of line cd are given by minus 2 minus of minus 6 that is 4 minus 8 minus of minus 4 that is minus 4 25 minus of minus 2 that is 28 so a2 is equal to 4 b2 is equal to minus 4 and c2 is equal to 28 and we know that the conditions for parallelism is a1 by a2 is equal to b1 by b2 is equal to c1 by c2 therefore we have a1 by a2 that is 2 by 4 which is equal to 1 by 2 b1 by b2 which is equal to minus 2 by minus 4 will be equal to 1 by 2 and c1 by c2 that is 14 by 28 is also equal to 1 by 2 now we have a1 by a2 is equal to b1 by b2 is equal to c1 by c2 therefore we can say that the line a b is parallel to the line cd this completes our session hope you enjoyed this session