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 YZ-plane 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 + (-5-4)2 + (8-7)2) = √(52 + (-9)2 + 12) = √(25 + 81 + 1) = √107

Step 3: Calculate the ratio in which the YZ-plane divides the line segment. Ratio = Distance from midpoint to (−2,4,7) / Total Distance = √((-2-0.5)2 + (4-(-0.5))2 + (7-7.5)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:

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

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

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

4. 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 = √((5-3)^2 + (4-2)^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 = √((9-3)^2 + (8-2)^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