 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that, find the angle between any two diagonals of a cube when origin lies at the center of a cube whose site is two units. We know that angle between any two lines is given by cos theta is equal to L1 L2 plus M1 M2 plus N1 N2, where L1 M1 N1 and L2 M2 N2 are the direction cosines of the two lines and direction cosines are given by the formula L is equal to plus minus A upon square root of A square plus B square plus C square. M is equal to plus minus B upon square root of A square plus B square plus C square. N is equal to plus minus C upon square root of A square plus B square plus C square. Science of direction cosines should be taken either all positive or all negative. And A, B, C are called direction ratios. That is, direction ratios of any line say P, Q are given by x2 minus x1, y2 minus y1, z2 minus z1 where x1, y1, z1 and x2, y2, z2 are the coordinates of point P and Q respectively. With this key idea let us proceed with the solution. Now we are given a cube A, B, C, D, E, F, G, H such that the side of the cube is 2 units and origin lies at the center of the cube. Since it is given that the side of the cube is 2 units and origin lies at the center of the cube taking O F center the origin divides the side of the cube into 2 equal halves that is 1 unit each. Therefore coordinates of its vertices are given by since A lies on the positive side of all the axis and has x, y and z magnitude equal to 1 the coordinates of A will be 1, 1, 1. Similarly since B lies on the positive side of x and z axis and on the negative side of y axis therefore coordinates of point B will be 1 minus 1, 1. It should be noted that points on the right side of the x axis are taken as positive and those on the left side of the x axis are taken as negative. Similarly points on the upward side of the y axis are taken as positive and those on the downward side are taken as negative and for z axis points on the outward side are taken as positive and those on the inward side are taken as negative. And this way we can find the vertices of the rest of the coordinates and thus we get point A with the coordinates 1, 1, 1, point B with the coordinates 1, minus 1, 1, point C with the coordinates minus 1, minus 1, 1, point D with the coordinates 1, minus 1, 1, point E with with the coordinates 1, 1, minus 1, point F with the coordinates 1, minus 1, minus 1, point G with the coordinates minus 1, minus 1, minus 1, point H with the coordinates minus 1, 1, minus 1. It is clear from the figure that B, H and C, E are the two diagonals of the cube and we need to find the angle between C, E and B, H. As we know the coordinates of point B, C, E and H, so we can find the direction ratios of line C, E and B, H. Using the key idea we get direction ratios of diagonal C, E are given by x2 minus x1 that is 1 minus of minus 1, y2 minus y1 that is 1 minus of minus 1, z2 minus z1 that is minus 1 minus 1 which can be written as 2, 2, minus 2. Similarly, direction ratios of diagonal B, H are given by, similarly, direction ratios of diagonal B, H are given by x2 minus x1 that is minus 1, minus 1, y2 minus y1 that is 1 minus of minus 1, z2 minus z1 that is minus 1 minus 1 which can be written as minus 2, 2, minus 2. Now we shall find direction cosines of diagonal C, E and B, H using the key idea which states that direction cosines are given by L is equal to plus minus A upon square root of A square plus B square plus C square, M is equal to plus minus B upon square root of A square plus B square plus C square, N is equal to plus minus C upon square root of A square plus B square plus C square where A, B, C E are the direction ratios of that line. Therefore, direction cosine of diagonal C E having direction ratios 2 to minus 2 that is A is equal to 2, B is equal to 2, C is equal to minus 2 are given by L1 is equal to A upon that is 2 upon square root of A square plus B square plus C square that is 2 square plus 2 square plus of minus 2 the whole square which is equal to 2 upon square root of 12. M1 is equal to B upon that is 2 upon square root of A square plus B square plus C square that is 2 square plus 2 square plus of minus 2 the whole square which is equal to 2 upon square root of 12. M1 is equal to C upon that is minus 2 upon square root of 2 square plus 2 square plus of minus 2 the whole square which is equal to minus 2 upon square root of 12. Similarly, we shall find direction cosines of diagonal B H having direction ratios minus 2 to minus 2 that is A is equal to minus 2, B is equal to 2, C is equal to minus 2 are given by L2 is equal to A upon that is minus 2 upon square root of A square plus B square plus C square that is minus 2 square plus 2 square plus of minus 2 the whole square which is equal to minus 2 upon square root of 12. M2 is given by B upon that is 2 upon square root of A square plus B square plus C square that is minus 2 square plus 2 square plus of minus 2 the whole square which is equal to 2 upon square root of 12. Similarly, N2 is equal to C upon square root of A square plus B square plus C square that is minus 2 upon square root of minus 2 square plus 2 square plus of minus 2 the whole square which is equal to minus 2 upon square root of 12. We need to find the angle between the two diagonals and from the key idea we know that angle between any two lines is given by cos theta is equal to L1L2 plus M1M2 plus N1N2 where L1M1N1 and L2M2N2 are the direction cosines of the two lines. Therefore, angle between and BH is given by cos theta is equal to L1L2 plus M1M2 plus N1N2 where L1M1N1 and L2M2N2 are the direction cosines of the diagonal CE and BH respectively or we can write cos theta is equal to L1L2 that is 2 upon square root of 12 into minus 2 upon square root of 12 plus M1M2 that is 2 upon square root of 12 into 2 upon square root of 12 plus N1N2 that is minus 2 upon square root of 12 into minus 2 upon square root of 12 or cos theta is equal to minus 4 upon 12 plus 4 upon 12 plus 4 upon 12 which gives the value of cos theta is equal to 4 upon 12 which further gives cos theta is equal to 1 by 3 which gives the value of theta as cos inverse of 1 by 3 hence angle between any two diagonals of a cube whose side is two units and origin lies at the center is given by cos inverse of 1 by 3 which is the required answer. This completes our session. Hope you enjoyed this session.