Solid Analytic Geometry

The following examples and case studies demonstrate how to apply the formulas and geometric properties of solid analytic geometry to solve spatial problems in 3D.

Distance and Midpoint in Space

Example 1: Calculating 3D Distance and Midpoint

Problem: Given two points in 3D space, P1(2,−1,4)P_1(2, -1, 4) and P2(5,3,−8)P_2(5, 3, -8), calculate the straight-line distance between them and determine the coordinates of their exact midpoint.

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Equations of Lines in 3D

Example 2: Finding Parametric and Symmetric Equations of a Line

Problem: Find the parametric and symmetric equations of a line passing through the points P1(−3,2,5)P_1(-3, 2, 5) and P2(1,4,−1)P_2(1, 4, -1).

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Equations of Planes in 3D

Example 3: Formulating the General Equation of a Plane

Problem: A geometric plane passes through the point P1(4,−1,3)P_1(4, -1, 3) and is perfectly perpendicular to a normal vector defined by direction numbers [2,5,−4][2, 5, -4]. Determine the general equation of this plane in the standard form Ax+By+Cz+D=0Ax + By + Cz + D = 0.

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Distance from a Point to a Plane

Example 4: Calculating Perpendicular Point-to-Plane Distance

Problem: Calculate the shortest absolute perpendicular distance from the point P(2,−3,6)P(2, -3, 6) directly to the infinitely spanning 3D plane described by the equation 3x−4y+12z−10=03x - 4y + 12z - 10 = 0.

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