 Hello and welcome to the session. In this session first we will discuss coordinate axis and coordinate planes in three-dimensional space. In three dimensions the coordinate axis of a rectangular Cartesian coordinate system are three mutually perpendicular lines. Let O be the origin and let O x, O y and O z be three mutually perpendicular lines. The line O x is the x-axis, O y is the y-axis and O z is the z-axis. These are the coordinate axis. The three planes determined by the pair of axis that is the planes x, o, y, y, o, z and z, o, x. These are the coordinate planes. This x, o, y plane is called the x, y plane then y, o, z plane is y, z plane and z, o, x plane is z, x plane. So these are the three coordinate planes. The three coordinate planes divide the space into eight parts called octants. Let's consider any point P in space. The coordinates of this point is always written in the form x, y, z. Here x is the distance of P from y, z plane then y is the distance of P from z, x plane and z is the distance of P from x, y plane. Coordinates of the origin O are always 0, 0, 0. If we take any point on x-axis it is of the form x, 0, 0 that is y and z coordinates are 0 then any point on y-axis would be of the form 0, y, 0 that is x and z coordinates are 0 then any point on z-axis is of the form 0, 0, z that is x and y coordinates are 0. Any point in y, z plane has coordinates of the form 0, y, z that is x coordinate would be 0 then any point in z, x plane would be of the form x, 0, z that is the y coordinate is 0 then any point in x, y plane is of the form x, y, 0 that is z coordinate would be 0. The sign of the coordinates of a point determine the octant in which the point lies. If the coordinates x, y, z are all positive then it lies in the first octant and if x is negative y is positive z is positive then it lies in the second octant, x negative, y negative and z positive it lies in the third octant, x positive, y negative, z positive it lies in the fourth octant, x positive, y positive, z negative it lies in the fifth octant then if x coordinate is negative y positive z negative it lies in the sixth octant then if x, y, z are all three negative then it lies in the seventh octant and if x is positive y, z are negative then it lies in the eighth octant. Consider a point t with coordinates 4, minus 2, 3. Now here x is positive, y is negative and z is positive so according to this table we can see that it lies in the fourth octant. So point p lies in the fourth octant. Next we discuss distance between two points. Suppose we are given two points p and q where the coordinates are point p v, x1, y1, z1 and the coordinates of the point q v, x2, y2, z2. The distance between these two points p and q is given by pq equal to square root of x2 minus x1 the whole square plus y2 minus y1 the whole square plus z2 minus z1 the whole square. Suppose we are given a point a with coordinates minus 2, 1, minus 3 and a point v with coordinates 4, 3, minus 6. Now the distance between the points a and b given by a, b is equal to square root of x2 minus x1 that is 4 minus of minus 2 the whole square plus y2 minus y1 the whole square plus z2 minus z1 the whole square that is equal to square root of 60 whole square plus 2 the whole square plus minus 3 the whole square that is equal to square root 49 and so we get a, b is equal to 7 units. Next we shall discuss section formula. Consider a point p with coordinates x1, y1, z1 and a point q with coordinates x2, y2, z2. Suppose that the point r divides the line segment joining the two points p and q that is r divides pq internally in the ratio m is to n then we have coordinates of point r are given by mx2 plus nx1 over m plus n, my2 plus ny1 upon m plus n then mz2 plus nz1 over m plus n and if the point r divides pq externally in the ratio m is to n then the coordinates of the point r are given by mx2 minus nx1 over m minus n then my2 minus ny1 upon mx2 minus ny1 upon mx2 minus nz2 minus nz1 by m minus n. Then the coordinates of the midpoint of the segment joining the points p and q that is of the line segment pq is given by x1 plus x2 by 2, y1 plus y2 by 2, z1 plus z2 by 2. If we have that the point r divides the segment pq in the ratio k is to 1 then the coordinates of the point r would be given by kx2 plus x1 by k plus 1 then ky2 plus y1 by k plus 1 then kz2 plus z1 by k plus 1. Consider a triangle ABC where the point a has coordinates x1, y1, z1, point b has coordinates x2, y2, z2 and the point b has coordinates x3, y3, z3. Then the coordinates of the centroid of the triangle ABC is given by x1 plus x2 plus x3 upon 3, y1 plus y2 plus y3 upon 3, z1 plus z2 plus z1 plus z2 plus z1 plus z2 plus z3 upon 3. Consider a point p with coordinates 5, 4, 2, point q with coordinates minus 1, minus 2, 4. Suppose the point r with coordinates x, y, z divides the line segment joining the points p and q that is the line segment pq internally in the ratio 2 is to 3 then we need to find the coordinates of the point r. Then from the formula x coordinate would be equal to mx2 now m is 2 into x2 that is minus 1 plus nx1 n is 3 and x1 is 5 upon m plus n that is 2 plus 3. Now this is equal to 13 upon 5 then dy coordinate is equal to my2 n is 2 y2 is minus 2 plus n y1 that is 3 into 4 and x1 is 5 upon m plus n that is 2 plus 3. Now this is equal to 13 upon 5 then dy coordinate is equal to my2 n is 2 y2 is minus 2 plus n y1 that is 3 into 4 upon m plus n that is 2 plus 3. This comes out to be equal to 8 upon 5 then we have the z coordinate would be equal to mz2 that is 2 into 4 plus n y1 that is 3 into 2 upon m plus n this is equal to 14 upon 5. So we get the coordinates of the point r are 13 upon 5 8 upon 5 14 upon 5. So this is how we use the section formula this completes the session hopefully you have understood the coordinate axis coordinate planes section formula distance between two points.