Answer :

Let A (x_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2}) 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

x_{1} = 4, y_{1} = 2, z_{1} = -6; x_{2} = 10, y_{2} = -16, z_{2} = 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 (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) 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

x_{1} = 4, y_{1} = 2, z_{1} = -6; x_{2} = 10, y_{2} = -16, z_{2} = 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 (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) 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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