First slide

Introduction to 3-D Geometry

Question

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

Moderate

A

(6, - 4, - 2), (8, - 10, 2)
B

(6, 4, - 2), (8, - 10, 2)
C

(6, - 4, - 2), (8, 10, 2)
D

None of the above

Solution

Let the point R₁, trisects the line PQ i.e., it divides the line in the ratio 1:2
$\begin{array}{l} \Rightarrow R_{1} = [\frac{1 \times 10 + 2 \times 4}{1 + 2}, \frac{1 \times (- 16) + 2 \times 2}{1 + 2}, \frac{1 \times 6 + 2 \times (- 6)}{1 + 2}] \\ = (\frac{10 + 8}{3}, \frac{- 16 + 4}{3}, \frac{6 - 12}{3}) = (\frac{18}{3}, \frac{- 12}{3}, \frac{- 6}{3}) \\ = (6, - 4, - 2) \end{array}$
Again, let the point R₂ divides PQ internally in the ratio 2: 1. Then,
$\begin{array}{l} \Rightarrow R_{2} = [\frac{2 \times 10 + 1 \times 4}{2 + 1}, \frac{2 \times (- 16) + 1 \times 2}{2 + 1}, \frac{2 \times 6 + 1 \times (- 6)}{1 + 2}] \\ = (\frac{20 + 4}{3}, \frac{- 32 + 2}{3}, \frac{12 - 6}{3}) = (\frac{24}{3}, \frac{- 30}{3}, \frac{6}{3}) \\ = (8, - 10, 2) \end{array}$
Hence, required points are (6,-4, - 2) and (8, - 10,2).

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