Find the coordinates of the point which divides the line segment joining the points (– 2, 3, 5) and (1, – 4, 6) in the ratio (i) 2 : 3 internally, (ii) 2 : 3 externally.
Let the line segment joining the points P (-2, 3, 5) and Q (1, -4, 6) be PQ.
(i) By Section Formula,
We know that the coordinates of the point R which divides the line segment joining two points P (x1, y1, z1) and Q (x2, y2, z2) internally in the ratio m : n is given by:
Comparing this information with the details given in the question, we have
x1 = -2, y1 = 3, z1 = 5; x2 = 1, y2 = -4, z2 = 6 and m = 2, n = 3
So,
The coordinates of the point which divides the line segment joining the points P (– 2, 3, 5) and Q (1, – 4, 6) in the ratio 2 : 3 internally is given by:
Hence, the coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) is 
.
(ii) By Section Formula,
We know that the coordinates of the point R which divides the line segment joining two points P (x1, y1, z1) and Q (x2, y2, z2) externally in the ratio m : n is given by:
Comparing this information with the details given in the question, we have
x1 = -2, y1 = 3, z1 = 5; x2 = 1, y2 = -4, z2 = 6 and m = 2, n = 3
So,
The coordinates of the point which divides the line segment joining the points P (– 2, 3, 5) and Q (1, – 4, 6) in the ratio 2 : 3 externally is given by:
Hence, the coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) is (-8, 17, 3).