Formula Sheet: Introduction to Three Dimensional Geometry

PQ is the distance between points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). This formula generalizes the distance formula from two-dimensional geometry to three dimensions, allowing us to find lengths in space.

The origin is the reference point in three-dimensional space, where all coordinates are zero. It serves as the starting point for measuring distances and directions.

The signs of x, y, z determine the octant location: 1st (+,+,+), 2nd (−,+,+), 3rd (−,−,+), 4th (+,−,+), 5th (+,+,−), 6th (−,+,−), 7th (−,−,−), 8th (+,−,−). Understanding octants aids in visualizing points in space.

A, B, C are the direction ratios of the normal vector, and x, y, z are the variables representing points on the plane. This equation describes flat surfaces in three-dimensional space.

For points P, Q, R to be collinear, the sum of the distances PQ and QR must equal distance PR. It is a fundamental criterion for checking alignment in three-dimensional geometry.

G is the centroid, the average point of the vertices A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃). Centroids are critical in various applications, including balancing and gravity analysis.

In a parallelogram, opposite sides are equal in length, ensuring stability and symmetry. This property is essential for solving geometrical problems and proofs.

Points on the axes have only one non-zero coordinate, simplifying many calculations in three-dimensional geometry.

This relation results in the locus of points forming the perpendicular bisector of segment AB, serving practical applications in design and navigation.

h, k, l are the center coordinates, and r is the radius. This formula describes all points equidistant from a center point, useful in modeling spherical shapes.

This is derived from the right triangle formed by dropping perpendiculars. It illustrates the Pythagorean theorem's application in three-dimensional space.

This defines the distance from point P(x, y, z) to point A(x₁, y₁, z₁) in three dimensions, offering insights into spatial relationships.

This equation relates to locus points and is used in geometric constructions where two points maintain equal distance from a third variable point.

This represents a sphere's equation in standard form, simplifying sphere properties for calculations and geometrical proofs.

This formula calculates the distance D from point (x, y, z) to a plane described by Ax + By + Cz + D = 0, important in optimization and spatial analysis.

This represents the direction ratios of a line in space. It helps in defining lines based on their slopes for analytical tasks and vector applications.

Where l is length, b is breadth, and h is height, this calculates the volume of three-dimensional shapes, relevant in real-world contexts such as packaging and space management.

This formula for angle θ between two lines with slopes m₁ and m₂ aids in understanding spatial orientations and relationships.

This equation helps locate all points that maintain equal distance to two distinct points, useful in geometric constructions.

This definition for determining the centroid of triangle ABC aids in analyzing properties of geometrical shapes effectively.