Page 5 - LESSON NOTE

7. Given that P (3, 2, – 4), Q (5, 4, – 6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides
PR..
Ans:- Let point Q (5, 4, –6) divide the line segment joining points P (3, 2, –4) and R (9, 8, –10) in the

ratio k : 1

Therefore, by section formula,

(5, 4, -6) = {(9 * k + 3)/(k + 1), (8 * k + 2)/(k + 1), (-10 * k - 4)/(k + 1)}

=> 5 = (9k + 3)/(k + 1)

=> 9k + 3 = 5(k + 1)

=> 9k + 3 = 5k + 5

=> 9k – 5k = 5 – 3

=> 4k = 2

=> k = 2/4

=> k = 1/2

Thus, point Q divides PR in the ratio = 1/2 : 1 = 1 : 2

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

Ans:- Let A and B be the points that trisect the line segment joining points P (4, 2, –6) and

Q(10, –16, 6).

Point A divides PQ in the ratio 1 : 2. Therefore, by section formula, the coordinates of point A

are given by

{(1 * 10 + 2 * 4)/(1 + 2), (-16 * 1 + 2 * 2)/(1 + 2), (1 * 6 + 2 * (-6))/(1 + 2)} = (6, -4, -2)

Point B divides PQ in the ratio 2 : 1. Therefore, by section formula, the coordinates of point B

are given by

{(2 * 10 + 1 * 4)/(2 + 1), (-16 * 2 + 2 * 1)/(2 + 1), (2 * 6 + 1 * (-6))/(2 + 1)} = (8, -10, 2)

Thus, (6, –4, –2) and (8, –10, 2) are the points that trisect the line segment joining points

P (4, 2, –6) and Q (10, –16, 6).