Chapter 12: Introduction to Three Dimensional Geometry Exercise 12.3

Let the line segment joining the points P (-2, 3, 5) and Q (1, -4, 6) be PQ.

(i) 2: 3 internally

By using the section formula,

We know that the coordinates of the point R, which divide the line segment joining two points, P (x₁, y₁, z₁) and Q (x₂, y₂, z₂), internally in the ratio m: n, are given by

Upon comparing, we have

x₁ = -2, y₁ = 3, z₁ = 5;

x₂ = 1, y₂ = -4, z₂ = 6 and

m = 2, n = 3

So, the coordinates of the point which divide the line segment joining the points P (– 2, 3, 5) and Q (1, – 4, 6) in the ratio 2 : 3 internally are given by

Hence, the coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) is (-4/5, 1/5, 27/5).

(ii) 2: 3 externally

By using the section formula,

We know that the coordinates of the point R, which divide the line segment joining two points, P (x₁, y₁, z₁) and Q (x₂, y₂, z₂), externally in the ratio m: n, are given by

Upon comparing, we have

x₁ = -2, y₁ = 3, z₁ = 5;

x₂ = 1, y₂ = -4, z₂ = 6 and

m = 2, n = 3

So, the coordinates of the point which divide the line segment joining the points P (– 2, 3, 5) and Q (1, – 4, 6) in the ratio 2: 3 externally are given by

∴ The coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) are (-8, 17, 3).

Let us consider Q divides PR in the ratio k: 1.

By using the section formula,

We know that the coordinates of the point R, which divides the line segment joining two points, P (x₁, y₁, z₁) and Q (x₂, y₂, z₂), internally in the ratio m : n, are given by

Upon comparing, we have

x₁ = 3, y₁ = 2, z₁ = -4;

x₂ = 9, y₂ = 8, z₂ = -10 and

m = k, n = 1

So, we have

9k + 3 = 5 (k+1)

9k + 3 = 5k + 5

9k – 5k = 5 – 3

4k = 2

k = 2/4

= ½

Hence, the ratio in which Q divides PR is 1: 2.

Let the line segment formed by joining the points P (-2, 4, 7) and Q (3, -5, 8) be PQ.

We know that any point on the YZ-plane is of the form (0, y, z).

So now, let R (0, y, z) divide the line segment PQ in the ratio k: 1.

Then,

Upon comparing, we have

x₁ = -2, y₁ = 4, z₁ = 7

x₂ = 3, y₂ = -5, z₂ = 8 and

m = k, n = 1

By using the section formula,

We know that the coordinates of the point R, which divide the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m: n are given by

So we have,

3k – 2 = 0

3k = 2

k = 2/3

Hence, the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8) is 2:3.

Let the point P divides AB in the ratio k: 1.

Upon comparing, we have

x₁ = 2, y₁ = -3, z₁ = 4;

x₂ = -1, y₂ = 2, z₂ = 1 and

m = k, n = 1

By using the section formula,

We know that the coordinates of the point R, which divide the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m : n are given by

So we have,

Now, we check if, for some value of k, the point coincides with point C.

Put (-k+2)/(k+1) = 0

-k + 2 = 0

k = 2

When k = 2, then (2k-3)/(k+1) = (2(2)-3)/(2+1)

= (4-3)/3

= 1/3

And, (k+4)/(k+1) = (2+4)/(2+1)

= 6/3

= 2

∴ C (0, 1/3, 2) is a point which divides AB in the ratio 2: 1 and is the same as P.

Hence, A, B, C are collinear.

Let A (x₁, y₁, z₁) and B (x₂, y₂, z₂) trisect the line segment joining the points P (4, 2, -6) and Q (10, -16, 6).

A divides the line segment PQ in the ratio 1: 2.

Upon comparing, we have

x₁ = 4, y₁ = 2, z₁ = -6

x₂ = 10, y₂ = -16, z₂ = 6 and

m = 1, n = 2

By using the section formula,

We know that the coordinates of the point R, which divide the line segment joining two points, P (x₁, y₁, z₁) and Q (x₂, y₂, z₂), internally in the ratio m : n, are given by

So we have,

Similarly, we know that B divides the line segment PQ in the ratio 2: 1.

Upon comparing, we have

x₁ = 4, y₁ = 2, z₁ = -6;

x₂ = 10, y₂ = -16, z₂ = 6 and

m = 2, n = 1

By using the section formula,

We know that the coordinates of the point R, which divide the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m: n are given by

So we have,

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

Exercise 12.1 Solutions: 4 Questions

Exercise 12.2 Solutions: 5 Questions

Miscellaneous Exercise on Chapter 12 Solutions: 6 Questions

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