Introduction to Three Dimensional Geometry - Section Formula

Position of a Point in 3D Space: A point in three-dimensional space is represented as $(x, y, z)$, which describes its distances from the $YZ$-plane, $ZX$-plane, and $XY$-plane respectively. Visually, think of a point $P$ as a vertex of a rectangular box whose three adjacent edges lie along the $X$, $Y$, and $Z$ axes.

Internal Division Concept: When a point $R$ lies on the line segment joining two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ such that it divides the segment in a given ratio $m:n$ internally. Visually, $R$ is located between $P$ and $Q$, and the ratio of the lengths $PR$ to $RQ$ is exactly $m/n$.

External Division Concept: When a point $R$ lies on the line passing through $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ but outside the segment $PQ$, it is said to divide $PQ$ externally in the ratio $m:n$. Visually, $R$ is placed on the extension of the line segment $PQ$, where the distance from $P$ to $R$ and $Q$ to $R$ maintains the ratio $m:n$.

Midpoint as a Special Case: The midpoint is a specific instance of internal division where the ratio $m:n$ is $1:1$. Visually, the midpoint $M$ is the unique point on the segment $PQ$ that is equidistant from both endpoints $P$ and $Q$.

Centroid of a Triangle: In 3D geometry, the centroid of a triangle with vertices $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, and $(x_3, y_3, z_3)$ is the point where the three medians intersect. Visually, it represents the geometric center or the 'balance point' of the triangular surface in space.

The Ratio $k:1$ Method: To find the ratio in which a point divides a line segment, it is often mathematically simpler to assume the ratio is $k:1$. If the resulting value of $k$ is positive, the division is internal; if $k$ is negative, the division is external.