 Hello and welcome to the session. In this session first we are going to discuss angle between two lines. We will now discuss the angle between two non-intersecting lines that is skew lines. The angle between two skew lines is the angle between two intersecting lines drawn from any point parallel to each of the given lines. Let L1, M1, M1 and L2, M2, M2 be the direction cosines of two given lines OP and OQ be the two lines the origin parallel to the given lines. Let theta be the angle between the lines. Here OP and OQ are the two lines through the origin parallel to the given lines and angle between the two lines that is OP and OQ is given by theta. Let OP be equal to R1 and OQ be equal to R2 that is let OP be equal to R1, OQ be equal to R2 be equal to B. We are now trying the ABC where AC is given by B, BC is equal to A and AB is equal to C. Then by cosine formula we say that cos of angle B is equal to A square plus C square minus of B square upon twice of A into C which implies that twice of A into C into cos of angle B is equal to A square plus C square minus of B square which implies that B square is equal to A square plus C square minus of twice of A into C into cos of angle B. Similarly we can find the values of A square and C square. Now here in trying the OPQ the OP is given by R1 and OQ is given by R2 and the value of PQ is given by B and theta is the angle between the lines OP and OQ. Then by using cosine formula we say that the value of D square is equal to R1 square plus R2 square minus of twice of R1 into R2 into cos of angle theta. Then by cosine formula we have D square is equal to R1 square plus R2 square minus of twice of R1 into R2 into cos of angle theta and mark this equation as 1. Let the coordinates of point P be x1, y1, z1 and point Q be x2, y2, z2 then OP square is given by R1 square is equal to x1 square plus y1 square plus z1 square that is the value of OP square is equal to R1 square. And by using distance formula the value of OP square will be equal to x1 minus 0D whole square plus y1 minus 0D whole square plus z1 minus 0D whole square which is equal to x1 square plus y1 square plus z1 square. Similarly OP square is equal to R2 square that is x2 square plus y2 square plus z2 square and the value of PQ square that is D square is given by x1 minus x2D whole square plus y1 minus y2D whole square plus z1 minus z2D whole square On solving further we get D square is equal to x1 square plus y1 square plus z1 square plus x2 square plus y2 square plus z2 square minus of 2 into x1 into x2 plus y1 into y2 plus z1 into z2. Therefore D square is equal to x1 square plus y1 square plus z1 square which is equal to R1 square plus x2 square plus y2 square plus z2 square which is equal to R2 square minus of 2 into x1 into x2 plus y1 into y2 plus z1 into z2 and mark this equation as 2. Now comparing equation 1 and equation 2 we get R1 into R2 Corsus theta is equal to x1 into x2 plus y1 into y2 plus z1 into z2 all we can say corsus theta is equal to x1 by r1 into x2 by r2 plus y1 by r1 into y2 by r2 plus z1 by r1 into z2 by r2. And we know that x1 is equal to l1 into r1 or we can also write it as x1 by r1 is equal to l1. Also x2 is equal to l2 into r2 which implies that x2 by r2 is equal to l2. Similarly we have y1 by r1 is equal to m1 and y2 by r2 is equal to m2 and z1 by r1 is equal to m1 and z2 by r2 is equal to n2. Therefore we can say that cos of angle theta is equal to x1 by r1 that is l1 into x2 by r2 that is l2 plus y1 by r1 that is m1 into y2 by r2 that is m2 plus z1 by r1 that is m1 into z2 by r2 that is m2 which can also be written as summation of l1 into l2. It gives the angle theta between the two lines. Now we shall find the angle theta in terms of direction ratio of the line. If direction cos of two lines l1, m1, m1 and l2, m2, m2 be proportional to b1, c1 and a2, b2, c2 then their actual values are plus minus s a1 upon square root of summation of a1 square plus minus s b1 upon square root of summation of a1 square plus minus s c1 upon square root of summation of a1 square and and plus minus of a2 upon square root of summation of a2 square plus minus of b2 upon square root of summation of a2 square plus minus of c2 upon square root of summation of a2 square respectively. then cos of theta is given by plus minus of summation of A1 into A2 upon square root of summation of A1 square into square root of summation of A2 square. Let us now discuss sine form and tangent form of the angle formula. We know that sine square theta plus cos square theta is equal to 1 which implies that sine square theta is equal to 1 minus cos square theta. Also we know that the value of cos of theta is equal to L1 into L2 plus M1 into M2 plus M1 into M2. We put this value of cos theta in the equation that is sine square theta is equal to 1 minus cos square theta. Therefore we get sine square theta is equal to 1 minus L1 into L2 plus M1 into M2 plus M1 into M2 the whole square. Now as we know that the value of L1 square plus M1 square plus M1 square is equal to 1 and L2 square plus M2 square plus M2 square is also equal to 1 which implies that L1 square plus M1 square plus M1 square into L2 square plus M2 square plus M2 square will be equal to 1 into 1 that is 1. Therefore, we can substitute this expression in place of 1, which implies that sine square theta is equal to L1 square plus N1 square plus N1 square into L2 square plus N2 square plus N2 square minus L1 into L2 plus N1 into N2 plus N1 into N2 V whole square, which implies that sine square theta is equal to L1 square into L2 square plus L1 square into N2 square plus L1 square into N2 square plus N1 square into L2 square plus N1 square into N2 square plus N1 square into N2 square plus N1 square into L2 square plus N1 square into N2 square plus n1 square into n2 square. Minus of l1 into l2 plus n1 into n2 plus n1 into n2 the whole square can be written as minus of l1 square into l2 square minus of n1 square into n2 square minus of n1 square into n2 square minus of twice of l1 into l2 into n1 into n2 minus of twice of n1 into n2 into n1 into n2 minus of twice of n1 into n2 into l1 into l2 Now we have time square of theta is equal to L1 squared into n2 squared plus L1 squared into n2 squared plus N1 squared into L2 squared plus n1 square into n2 square plus n1 square into l2 square plus n1 square into n2 square minus of twice of l1 into l2 into n1 into n2 minus of twice of n1 into n2 into n1 into minus of twice of n1 into n2 into l1 into l2 which implies that sin square of theta is equal to l1 square into m2 square plus n1 square into l2 square minus of twice of l1 into l2 into n1 into n2 plus l1 square into n2 square plus n1 square into l2 square minus of twice of l1 into l2 into n1 into l2 plus n1 square into n2 square plus n1 square into n2 square minus of twice of n1 into n2 into n1 into n2 which implies that sin square theta can be written as l1 into n2 minus of n1 into l2 the whole square plus l1 into n2 minus of n1 into l2 the whole square plus n1 into n2 minus of n1 into n2 the whole square therefore sin square theta is equal to summation of l1 into n2 minus of n1 into l2 the whole square which implies that sin of theta is equal to plus minus of square root of summation of l1 into n2 minus of n1 into l2 the whole square and we know that tan of angle theta is equal to sin of theta by cos of theta which is equal to plus minus of square root of summation of l1 into n2 minus of n1 into l2 the whole square cos theta which is equal to summation of l1 into l2 if the direction cos of two lines be proportional to a1 b1 c1 and a2 b2 c2 then their actual values are plus minus a1 upon square root of summation of a1 square plus minus b1 upon square root of summation of a1 square plus minus c1 upon square root of summation of a1 square and plus minus a2 upon square root of summation of a2 square plus minus b2 upon square root of summation of a2 square plus minus c2 upon square root of summation of a2 square therefore cos of angle theta 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 therefore put l1 is equal to a1 upon square root of summation of a1 square l2 is equal to a2 upon square root of summation of a2 square m1 is equal to b1 upon square root of summation of a1 square m2 is equal to b2 upon square root of summation of a2 square m1 is equal to c1 upon square root of summation of a1 square and m2 is equal to c2 upon square root of summation of a2 square and sin square theta is equal to l1 into m2 minus of l2 into m1 the whole square plus m1 into m2 minus of m2 into m1 the whole square plus m1 into l2 minus of m2 into l1 the whole square can now be written as sin square theta is equal to l1 into m2 that is a1 upon square root of summation of a1 square into b2 upon square root of summation of a2 square minus of l2 into m1 that is a2 upon square root of summation of a2 square into b1 upon square root of summation of a1 square b whole square plus m1 into m2 that is b1 upon square root of summation of a1 square into c2 upon square root of summation of a2 square minus m2 into m1 that is b2 upon square root of summation of a2 square into c1 upon square root of summation of a1 square the whole square plus m1 into l2 that is c1 upon square root of summation of a1 square into a2 upon square root of summation of a2 square minus of m2 into l1 that is c2 upon square root of summation of a2 square into a1 upon square root of summation of a1 square the whole square therefore sin square theta is equal to a1 into b2 minus of a2 into b1 the whole square upon summation of a1 square into summation of a2 square plus b1 into c2 minus of b2 into c1 the whole square upon summation of a1 square into summation of a2 square plus c1 into a2 minus of c2 into a1 the whole square upon summation of a1 square into summation of a2 square which implies that sin of angle theta is equal to plus minus of square root of into b2 minus of a2 into b1 the whole square plus b1 into c2 minus of b2 into c1 the whole square plus c1 into a2 minus of c2 into a1 the whole square whole upon square root of summation of a1 square into square root of summation of a2 square and cos of theta is given by plus minus of a1 into a2 plus b1 into b2 plus c1 into c2 whole upon square root of summation of a1 square into square root of summation of a2 square tan of theta is equal to sin of theta by cos of theta which is equal to square root of a1 into b2 minus of a2 into b1 the whole square plus b1 into c2 minus of b2 into c1 the whole square plus c1 into a2 minus of c2 into a1 the whole square upon a1 into a2 plus b1 into b2 plus c1 into c2 now let us write a1 into b2 minus of a2 into b1 the whole square plus b1 into c2 minus of b2 into c1 the whole square plus c1 into a2 minus of c2 into a1 the whole square as summation of a1 into b2 minus of a2 into b1 the whole square to a2 plus b1 into b2 plus c1 into c2 as summation of a1 into a2 then tan of angle theta can be written as square root of summation of a1 into b2 minus of a2 into b1 the whole square upon summation of a1 into a2 this completes our session hope you enjoyed this session