Introduction to Three Dimensional Geometry Exercise 3
Question:
Find the coordinates of the point which divides the line segment joining the points (−2,3,5) and (1,−4,6) in the ratio (i) 2:3 internally (ii) 2:3 externally.
Answer:
(i) Internally:
Let the coordinates of the point be (x, y, z).
Then, according to the given condition,
\frac{\overrightarrow{PQ}}{\overrightarrow{PR}} = \frac{2}{3}
Where PQ and PR are the vectors joining the points (−2,3,5) and (1,−4,6) to the point (x,y,z) respectively.
Therefore,
\overrightarrow{PQ} = 2\overrightarrow{PR}
⇒ (x + 2)^2 + (y  3)^2 + (z  5)^2 = 4(x  1)^2 + (y + 4)^2 + (z  6)^2
⇒ (x + 2)^2 + (y  3)^2 + (z  5)^2 = 4(x^2  2x + 1) + (y^2 + 8y + 16) + (z^2  12z + 36)
⇒ x^2 + 4x + 4 + y^2  6y + 9 + z^2  10z + 25 = 4x^2  8x + 4 + y^2 + 8y + 16 + z^2  12z + 36
⇒ 3x^2  4x + y^2  6y + z^2  10z + 9 = 0
⇒ x^2 + 4x + y^2  6y + z^2  10z = 9
⇒ x^2 + 4x + 4 + y^2  6y + 9 + z^2  10z + 25 = 9 + 25
⇒ x^2 + 4x + y^2  6y + z^2  10z = 16
⇒ x^2 + 4x + 16 + y^2  6y + 36 + z^2  10z = 16 + 36
⇒ x^2 + 4x + 16 + y^2  6y + 36 + z^2  10z = 52
Comparing the coefficients of x^2, y^2 and z^2, we get
x = 2, y = 1 and z = 4
Therefore, the coordinates of the point which divides the line segment joining the points (−2,3,5) and (1,−4,6) internally in the ratio 2:3 is (−2, 1, 4).
(ii) Externally:
Let the coordinates of the point be (x, y, z).
Then, according to the given condition,
\frac{\overrightarrow{PQ}}{\overrightarrow{PR}} = \frac{2}{3}
Where PQ and PR are the vectors joining the points (−2,3,5) and (1,−4,6) to the point (x,y,z) respectively.
Therefore,
\overrightarrow{PQ} = \frac{2}{3}\overrightarrow{PR}
⇒ (x + 2)^2 + (y  3)^2 + (z  5)^2 = \frac{4}{9}(x  1)^2 + \frac{1}{9}(y + 4)^2 + \frac{1}{9}(z  6)^2
⇒ (x + 2)^2 + (y  3)^2 + (z  5)^2 = \frac{4}{9}(x^2  2x + 1) + \frac{1}{9}(y^2 + 8y + 16) + \frac{1}{9}(z^2  12z + 36)
⇒ x^2 + 4x + 4 + y^2  6y + 9 + z^2  10z + 25 = \frac{4}{9}(x^2  2x + 1) + \frac{1}{9}(y^2 + 8y + 16) + \frac{1}{9}(z^2  12z + 36)
⇒ \frac{9}{4}x^2 + 6x + \frac{9}{4} + \frac{9}{4}y^2  \frac{9}{2}y + \frac{9}{4}z^2  \
Question:
Using section formula show that the points A (2,−3,4),B (−1,2,1) and C(0,1/3,2) are collinear.
Answer:
Step 1: Find the length of AB
AB = √((−1−2)2 + (2−(−3))2 + (1−4)2 )
= √(32 + 52 + (−3)2 )
= √(9 + 25 + 9)
= √43
Step 2: Find the length of BC
BC = √((0−(−1))2 + (1/3−2)2 + (2−1)2 )
= √(12 + (−5/3)2 + 12 )
= √(1 + 25/9 + 1)
= √(27/9)
Step 3: Find the length of AC
AC = √((2−0)2 + (−3−1/3)2 + (4−2)2 )
= √(22 + (−19/3)2 + 22 )
= √(4 + 361/9 + 4)
= √(369/9)
Step 4: Apply Section Formula
AB/AC = BC/BC
√43/√(369/9) = √(27/9)/√(27/9)
43/369 = 27/27
1 = 1
Hence, the points A (2,−3,4),B (−1,2,1) and C(0,1/3,2) are collinear.
Question:
Find the ratio in which the YZplane divides the line segment formed by joining the points (−2,4,7) and (3,−5,8).
Answer:
Step 1: Calculate the midpoint of the line segment. Midpoint = ( ( 2 + 3 )/2 , ( 4 + 5 )/2 , ( 7 + 8 )/2 ) = ( (1)/2 , (1)/2 , (15)/2 ) = ( 0.5 , 0.5 , 7.5 )
Step 2: Calculate the distance between the two points. Distance = √((3(2))2 + (54)2 + (87)2) = √(52 + (9)2 + 12) = √(25 + 81 + 1) = √107
Step 3: Calculate the ratio in which the YZplane divides the line segment. Ratio = Distance from midpoint to (−2,4,7) / Total Distance = √((20.5)2 + (4(0.5))2 + (77.5)2) / √107 = √(2.52 + 4.52 + (0.5)2) / √107 = √(6.25 + 20.25 + 0.25) / √107 = √(26.75) / √107 = 5.2 / 10.3 = 0.5048543689320389
Question:
Find the coordinates of the points which trisect the line segment joining the points P(4,2,−6) and Q(10,−16,6)
Answer:

Calculate the midpoint of the line segment joining the points P and Q by using the formula: Midpoint = (P + Q)/2

Substitute the given coordinates in the formula: Midpoint = ((4,2,6) + (10,16,6))/2

Simplify the equation to get the coordinates of the midpoint: Midpoint = (14,7,0)/2

Divide each coordinate by 2 to get the coordinates of the midpoint: Midpoint = (7,3.5,0)
Therefore, the coordinates of the points which trisect the line segment joining the points P(4,2,6) and Q(10,16,6) are (7,3.5,0).
Question:
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.
Answer:
Step 1: Calculate the distance between points P and Q. Distance between P and Q = √((53)^2 + (42)^2 + (6(4))^2) = √(4 + 4 + 4) = √12 = 2√3
Step 2: Calculate the distance between points P and R. Distance between P and R = √((93)^2 + (82)^2 + (10(4))^2) = √(36 + 36 + 36) = √108 = 6√3
Step 3: Calculate the ratio in which Q divides PR. Ratio in which Q divides PR = Distance between P and Q / Distance between P and R = 2√3 / 6√3 = 1/3