Find the coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6).
Let A (x1, y1, z1) and B (x2, y2, z2) trisect the line segment joining the points P (4, 2, -6) and Q (10, -16, 6).
⇒ A divides the line segment PQ in the ratio 1 : 2.
Then,
Comparing this information with the details given in the question, we have
x1 = 4, y1 = 2, z1 = -6; x2 = 10, y2 = -16, z2 = 6 and m = 1, n = 2
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:
So, we have
The coordinates of A = 
Similarly, We know that B divides the line segment PQ in the ratio 2 : 1.
Then,
Comparing this information with the details given in the question, we have
x1 = 4, y1 = 2, z1 = -6; x2 = 10, y2 = -16, z2 = 6 and m = 2, n = 1
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:
So, we have
The coordinates of B = 
Hence, the coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6) are (6, -4, -2) and (8, -10, 2).
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