Section Formula in 3D Geometry

The section formula is a mathematical expression used to find the coordinates of a point that divides a line segment into two parts in a given ratio. This formula is applicable in both two-dimensional (2D) and three-dimensional (3D) geometry. In 3D geometry, the formula helps us to find the coordinates of a point in space.

Internal Division

When a point divides a line segment internally in a given ratio, we use the internal division formula. Let's consider a line segment with endpoints $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$. If a point $P(x, y, z)$ divides the line segment $AB$ in the ratio $m:n$, then the coordinates of $P$ are given by:

$$ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}, \quad z = \frac{mz_2 + nz_1}{m + n} $$

External Division

For the case of external division, where a point divides the line segment externally in a given ratio, the formula is slightly different. If a point $P(x, y, z)$ divides the line segment $AB$ externally in the ratio $m:n$, then the coordinates of $P$ are given by:

$$ x = \frac{mx_2 - nx_1}{m - n}, \quad y = \frac{my_2 - ny_1}{m - n}, \quad z = \frac{mz_2 - nz_1}{m - n} $$

Midpoint Formula

A special case of the section formula is when a point divides a line segment into two equal parts, i.e., the midpoint. The midpoint $M(x, y, z)$ of a line segment with endpoints $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ has the coordinates:

$$ x = \frac{x_1 + x_2}{2}, \quad y = \frac{y_1 + y_2}{2}, \quad z = \frac{z_1 + z_2}{2} $$

Differences and Important Points

Here is a table summarizing the differences and important points of the section formula:

Aspect	Internal Division	External Division	Midpoint
Ratio	Divides line segment in a given ratio $m:n$	Divides line segment in a given ratio $m:n$	Divides line segment into equal halves
Formula for $x$	$\frac{mx_2 + nx_1}{m + n}$	$\frac{mx_2 - nx_1}{m - n}$	$\frac{x_1 + x_2}{2}$
Formula for $y$	$\frac{my_2 + ny_1}{m + n}$	$\frac{my_2 - ny_1}{m - n}$	$\frac{y_1 + y_2}{2}$
Formula for $z$	$\frac{mz_2 + nz_1}{m + n}$	$\frac{mz_2 - nz_1}{m - n}$	$\frac{z_1 + z_2}{2}$
Nature of Division	The point lies between $A$ and $B$	The point lies outside the segment $AB$	The point is exactly in the middle

Examples

Example 1: Internal Division

Find the coordinates of the point which divides the line segment joining the points $A(2, -3, 5)$ and $B(-1, 4, 6)$ in the ratio $2:3$ internally.

Solution:

Using the internal division formula:

$$ x = \frac{2(-1) + 3(2)}{2 + 3} = \frac{4}{5}, \quad y = \frac{2(4) + 3(-3)}{2 + 3} = \frac{2}{5}, \quad z = \frac{2(6) + 3(5)}{2 + 3} = \frac{27}{5} $$

So, the coordinates of the point are $\left(\frac{4}{5}, \frac{2}{5}, \frac{27}{5}\right)$.

Example 2: External Division

Find the coordinates of the point which divides the line segment joining the points $A(1, 2, 3)$ and $B(4, 5, 6)$ in the ratio $1:2$ externally.

Solution:

Using the external division formula:

$$ x = \frac{1(4) - 2(1)}{1 - 2} = \frac{2}{-1} = -2, \quad y = \frac{1(5) - 2(2)}{1 - 2} = \frac{1}{-1} = -1, \quad z = \frac{1(6) - 2(3)}{1 - 2} = \frac{0}{-1} = 0 $$

So, the coordinates of the point are $(-2, -1, 0)$.

Example 3: Midpoint

Find the midpoint of the line segment joining the points $A(3, -1, 4)$ and $B(7, 6, 2)$.

Solution:

Using the midpoint formula:

$$ x = \frac{3 + 7}{2} = 5, \quad y = \frac{-1 + 6}{2} = \frac{5}{2}, \quad z = \frac{4 + 2}{2} = 3 $$

So, the midpoint is $(5, \frac{5}{2}, 3)$.

Understanding the section formula and its applications is crucial for solving problems in coordinate geometry, especially in 3D space. It is widely used in various fields such as physics, engineering, and computer graphics.