 Hello and welcome to the session. My name is Asha and I am going to help you with the following question that says find the coordinates of the point which divides the line segment joining the points minus 2, 3, 5 and 1 minus 4, 6 in the ratio first as 2 is for 3 internally and second is 2 is for 3 externally. So first let us learn the section formula. It says F, P with coordinates x1, y1 and z1 and Q with coordinates x2, y2 and z2 are two points then the coordinates of the point which divides the line segment PQ internally in the ratio m is to n, m is nx1 upon m plus n, my2 plus n by 1 upon m plus n and mz2 plus nz1 upon m plus n. The coordinates which divides the line segment in the ratio m is to n are given by mx2 minus nx1 upon m minus n, my2 minus n by 1 upon m minus n and mz2 minus nz1 upon m minus n. So with the help of these two ideas we are going to solve the above problem so this is our key idea. Let us now start with the solution and name the two points a and b. So a has coordinates minus 2, 3 and 5 and b has coordinates 1 minus 4 and 6. Now first we have to find the coordinates of the point which divides the line segment in the ratio 2 is to 3 internally. So let P with coordinates x, y and z be the required point which divides the line segment joining a and b in the ratio 2 is to 3 internally. Now by the key idea here x1 is equal to minus 2, y1 is equal to 3 and z1 is equal to 5 and x2 is equal to 1, y2 is equal to minus 4 and z2 is equal to 6. Also m is equal to 2 and n is equal to 3. So we shall be applying the section formula to find the coordinates of the point P which divides the line segment a, b and the ratio 2 is to 3 internally. So x, y, z are given by, the x coordinate is given by m into x2 plus n into x1 upon m plus n. So m is 2 into 1 plus n into x1 upon m plus n 2 plus 3, y coordinate is given by m into y2 plus n into y1 upon m plus n. So we have 2 into minus 4 plus 3 into 3 upon 2 plus 3 and the z coordinate is given by m into z2 plus n which is 3 into z1 upon m plus n. So this is further equal to minus 4 upon 5, 1 upon 5, 27 upon 5. Hence the coordinates of the point are minus 4 upon 5, 1 upon 5 and 27 upon 5. Thus the required point is minus 4 upon 5, 1 upon 5 and 27 upon 5. Now let's proceed on to the second part which says 2 is to 3 externally. Here the values of x1, y1 and z1 and x2, y2 and z2 and m and n are same as in the first part. Just here we have to find that the point P, the coordinates of the point P which cuts the line segment a, b and the ratio 2 is to 3 externally. So its coordinates x, y, z are given by, x is given by m into x2 minus n into x1. So m is 2, x2 is 1 minus 3 into minus 2 upon 2 minus 3, y is given by 2 into minus 4 minus 3 into 3 upon 2 minus 3 and z is given by 2 into 6 minus 3 into 5 upon 2 minus 3. And I'm simplifying, this is further equal to minus 8, 17 and 3. Hence the answer is the point is minus 8, 17 and 3. So this completes the second part also. Hope you have understood it. Take care and have a good day.