CBSE Class 11 Maths: Introduction to 3D Geometry - Exercise 12.3

Understanding the Section Formula in Three Dimensions

In two-dimensional geometry, you learned about the section formula to find a point dividing a line segment. The principles extend seamlessly into three dimensions, where each point now has three coordinates (x, y, z).

Consider two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in 3D space. If a point R(x, y, z) divides the line segment PQ in the ratio m:n, we have two primary cases:

1. Internal Division: When the point R lies between P and Q. The coordinates of R are given by:
$R = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)$

This formula is derived by extending the 2D concept to the z-coordinate. Imagine projecting the line segment and the dividing point onto the x-axis, y-axis, and z-axis separately. The 2D section formula applies to each projection. For example, on the x-axis, the point dividing x₁ and x₂ in ratio m:n is (mx₂ + nx₁)/(m+n).

2. External Division: When the point R lies outside the line segment PQ, either on the extension of PQ beyond Q or beyond P. In this case, the ratio is still m:n, but one of the segments (either PR or RQ) is considered negative in direction relative to the other. The coordinates of R are given by:
$R = \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}, \frac{mz_2 - nz_1}{m-n} \right)$

Notice the change in the sign in both the numerator and the denominator for external division. It's crucial to correctly identify whether the division is internal or external, as it dictates the sign in the formula. A common way to think about it is that for internal division, the ratio is positive (m:n), and for external division, it's negative (m:(-n) or (-m):n).

Special Cases: Midpoint and Centroid Formulae

Midpoint Formula (3D)

The midpoint is a special case of internal division where the ratio m:n is 1:1. That is, the point divides the segment into two equal halves. Substituting m=1 and n=1 into the internal section formula, the coordinates of the midpoint M of a line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are given by: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)$

Centroid of a Triangle (3D)

The centroid of a triangle is the point of intersection of its medians. For a triangle with vertices A(x₁, y₁, z₁), B(x₂, y₂, z₂), and C(x₃, y₃, z₃), the coordinates of the centroid G are the average of the coordinates of its vertices. This is because the median divides the opposite side in a 1:1 ratio, and the centroid divides each median in a 2:1 ratio. The formula is: $G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right)$

Applying the Section Formula: Worked Examples

• Example 1: Internal Division Find the coordinates of the point that divides the line segment joining the points P(2, -1, 4) and Q(4, 3, -2) internally in the ratio 2:3. Step 1: Identify the given points and ratio. Points: P(x₁, y₁, z₁) = (2, -1, 4) and Q(x₂, y₂, z₂) = (4, 3, -2). Ratio: m:n = 2:3 (internal division). Step 2: Apply the internal section formula. $x = \frac{mx_2 + nx_1}{m+n} = \frac{2(4) + 3(2)}{2+3} = \frac{8 + 6}{5} = \frac{14}{5}$ $y = \frac{my_2 + ny_1}{m+n} = \frac{2(3) + 3(-1)}{2+3} = \frac{6 - 3}{5} = \frac{3}{5}$ $z = \frac{mz_2 + nz_1}{m+n} = \frac{2(-2) + 3(4)}{2+3} = \frac{-4 + 12}{5} = \frac{8}{5}$ Step 3: State the coordinates of the dividing point. The coordinates of the point are $(\frac{14}{5}, \frac{3}{5}, \frac{8}{5})$.
• Example 2: External Division Find the coordinates of the point that divides the line segment joining R(3, 2, -1) and S(1, -2, 5) externally in the ratio 3:2. Step 1: Identify the given points and ratio. Points: R(x₁, y₁, z₁) = (3, 2, -1) and S(x₂, y₂, z₂) = (1, -2, 5). Ratio: m:n = 3:2 (external division). Step 2: Apply the external section formula. $x = \frac{mx_2 - nx_1}{m-n} = \frac{3(1) - 2(3)}{3-2} = \frac{3 - 6}{1} = -3$ $y = \frac{my_2 - ny_1}{m-n} = \frac{3(-2) - 2(2)}{3-2} = \frac{-6 - 4}{1} = -10$ $z = \frac{mz_2 - nz_1}{m-n} = \frac{3(5) - 2(-1)}{3-2} = \frac{15 + 2}{1} = 17$ Step 3: State the coordinates of the dividing point. The coordinates of the point are $(-3, -10, 17)$.
• Example 3: Finding the Centroid of a Triangle Calculate the centroid of a triangle whose vertices are A(1, 2, 3), B(2, -1, 4), and C(-1, 3, 2). Step 1: Identify the coordinates of the vertices. Vertices: A(x₁, y₁, z₁) = (1, 2, 3), B(x₂, y₂, z₂) = (2, -1, 4), C(x₃, y₃, z₃) = (-1, 3, 2). Step 2: Apply the centroid formula. $G_x = \frac{x_1 + x_2 + x_3}{3} = \frac{1 + 2 + (-1)}{3} = \frac{2}{3}$ $G_y = \frac{y_1 + y_2 + y_3}{3} = \frac{2 + (-1) + 3}{3} = \frac{4}{3}$ $G_z = \frac{z_1 + z_2 + z_3}{3} = \frac{3 + 4 + 2}{3} = \frac{9}{3} = 3$ Step 3: State the coordinates of the centroid. The centroid of the triangle is $(\frac{2}{3}, \frac{4}{3}, 3)$.

Avoiding Common Pitfalls in 3D Section Formula

When applying the section formula, students often make a few common errors. Be mindful of these to secure full marks:

1. Mixing Internal and External Division: The most frequent mistake is using the wrong formula (e.g., using '-' for internal or '+' for external). Always double-check if the problem states 'internally' or 'externally' and apply the corresponding formula with correct signs.
2. Incorrect Ratio Order (m:n vs n:m): Ensure you consistently map 'm' to the coordinates of the second point (x₂, y₂, z₂) and 'n' to the first point (x₁, y₁, z₁). Swapping them will lead to incorrect results.
3. Coordinate Swapping: Be careful not to mix up x, y, and z coordinates for each point. For example, use x₁ with x₂, y₁ with y₂, and z₁ with z₂. It's easy to accidentally pair x₁ with y₂ if not careful.
4. Algebraic Errors: The formulas involve basic arithmetic (addition, subtraction, multiplication, division). Take your time with calculations, especially with negative numbers, to avoid silly mistakes.

Practice Questions with Solutions

• Q: Find the coordinates of the point that divides the line segment joining the points A(-2, 3, 5) and B(1, -4, 6) internally in the ratio 2:3. A: Step 1: Identify points and ratio. P(x₁, y₁, z₁) = (-2, 3, 5), Q(x₂, y₂, z₂) = (1, -4, 6), m=2, n=3. Step 2: Apply internal section formula for x, y, z. x = (21 + 3(-2))/(2+3) = (2 - 6)/5 = -4/5 y = (2(-4) + 33)/(2+3) = (-8 + 9)/5 = 1/5 z = (26 + 35)/(2+3) = (12 + 15)/5 = 27/5 Final answer: The coordinates of the point are (-4/5, 1/5, 27/5).
• Q: Determine the coordinates of the point which divides the line segment joining the points (1, -2, 3) and (3, 4, -5) externally in the ratio 4:3. A: Step 1: Identify points and ratio. P(x₁, y₁, z₁) = (1, -2, 3), Q(x₂, y₂, z₂) = (3, 4, -5), m=4, n=3. Step 2: Apply external section formula for x, y, z. x = (43 - 31)/(4-3) = (12 - 3)/1 = 9 y = (44 - 3(-2))/(4-3) = (16 + 6)/1 = 22 z = (4(-5) - 33)/(4-3) = (-20 - 9)/1 = -29 Final answer: The coordinates of the point are (9, 22, -29).
• Q: If the midpoint of the line segment joining (3, 4, 5) and (x, y, z) is (0, 0, 0), find the values of x, y, and z. A: Step 1: Use the midpoint formula. Midpoint (M) = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). Given M = (0, 0, 0), P₁ = (3, 4, 5), P₂ = (x, y, z). Step 2: Equate coordinates and solve for x, y, z. (3 + x)/2 = 0 => 3 + x = 0 => x = -3 (4 + y)/2 = 0 => 4 + y = 0 => y = -4 (5 + z)/2 = 0 => 5 + z = 0 => z = -5 Final answer: The values are x = -3, y = -4, z = -5.
• Q: The vertices of a triangle are A(2, -3, 1), B(-1, 0, 4), and C(5, 2, -2). Find the coordinates of its centroid. A: Step 1: Identify the coordinates of the vertices. A(x₁, y₁, z₁) = (2, -3, 1), B(x₂, y₂, z₂) = (-1, 0, 4), C(x₃, y₃, z₃) = (5, 2, -2). Step 2: Apply the centroid formula. G_x = (2 + (-1) + 5)/3 = (2 - 1 + 5)/3 = 6/3 = 2 G_y = (-3 + 0 + 2)/3 = -1/3 G_z = (1 + 4 + (-2))/3 = (1 + 4 - 2)/3 = 3/3 = 1 Final answer: The centroid of the triangle is (2, -1/3, 1).

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