Analytic Geometry: Distances and Angles in Space
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Analytic Geometry: Distances and Angles
Distance Between a Point and a Line
The distance of a point, P, to a line, r, is the smallest distance from the point to the infinite points on the line.
This distance corresponds to the perpendicular from the point to the line.
Distance Between Parallel Lines
The distance of a line, r, to another parallel line, s, is the distance from any point on r to s.
Distance Between Intersecting Lines
The distance between two intersecting lines is measured along the common perpendicular.
Let 
and 
be the direction vectors of the lines r and s.
Vectors 
determine a parallelepiped whose height is the distance between the two lines.
The volume of a parallelepiped is 
.
Given that the volume is the absolute value of the scalar triple product of three vectors and the area of the base is the magnitude of the cross product of the direction vectors of the lines, the height (the distance between the two lines) is equal to:
Distance From a Point to a Plane
The distance of a point, P, to a plane, $\pi$, is the smallest distance from the point to the infinite points of the plane.
This corresponds to the distance from the point perpendicular to the plane.
Distance Between Parallel Planes
To calculate the distance between two parallel planes, find the distance of any point on one plane to the other.
This can also be calculated using their equations:
Angle Between Two Lines
The angle between two lines is equal to the acute angle determined by their direction vectors.
Two lines are perpendicular if their direction vectors are orthogonal.
Angle Between Two Planes
The angle between two planes is equal to the acute angle determined by the normal vectors of these planes.
Two planes are perpendicular if their normal vectors are orthogonal.
Angle Between a Line and a Plane
The angle between a line and a plane is equal to the complementary acute angle formed by the line's direction vector and the plane's normal vector.
If line r is perpendicular to plane $\pi$, the direction vector of line r and the normal vector of the plane are parallel, meaning their components are proportional.
The angle formed by a line, r, and a plane, $\pi$, is the angle formed by r with its orthogonal projection onto $\pi$, denoted r'.
Area Calculations
Area of a Triangle 
and Parallelogram Area 
Geometrically, the magnitude of the vector product of two vectors equals the area of the parallelogram whose sides are defined by these vectors.
Volume Calculations
Volume of a Tetrahedron
The volume of a tetrahedron is equal to 1/6 of the absolute value of the scalar triple product.
Volume of a Parallelepiped
Geometrically, the absolute value of the scalar triple product represents the volume of the parallelepiped whose edges are three vectors originating from a single vertex.