Question

Find the coordinates of the points which tisect the line segment joining the points P(4, 2, –6) and Q(10, –16, 6). 

Answer 1

Let P\[\equiv\](4, 2,\[-\]6) and Q\[\equiv\](10,\[-\]16, 6) 
16, 6)
Let A and B be the point of trisection.
Then, we have:
PA = AB = BQ
∴ PA:AQ = 1:2
Thus, A divides PQ internally in the ratio 1:2.
∴ A\[\equiv\]\[\left( \frac{1 \times 10 + 2 \times 4}{1 + 2}, \frac{1 \times \left( - 16 \right) + 2 \times 2}{1 + 2}, \frac{1 \times 6 + 2 \times \left( - 6 \right)}{1 + 2} \right)\]\[\equiv \left( \frac{10 + 8}{3}, \frac{- 16 + 4}{3}, \frac{6 - 12}{3} \right)\]
\[ \equiv \left( \frac{18}{3}, \frac{- 12}{3}, \frac{- 6}{3} \right)\]
\[ \equiv \left( 6, - 4, - 2 \right)\]

\[\Rightarrow\]A\[\equiv\](6,\[-\]4,\[-\]AB = BQ       

Therefore, B is the mid-point of AQ.
∴ B\[\equiv\] \[\left( \frac{6 + 10}{2}, \frac{- 4 - 16}{2}, \frac{- 2 + 6}{2} \right)\]

\[\equiv \left( \frac{16}{2}, \frac{- 20}{2}, \frac{4}{2} \right)\]
\[ \equiv \left( 8, - 10, 2 \right)\]

\[\Rightarrow\]B \[\equiv\](8,\[-\]10, 2)