The following is a brief overview of some common concepts regarding vector-based math for lines and planes in 3-D space:
Direction Angles/Cosines
Unit Vector
Dot Product
Vector Projection
Scalar Projection
Cross Product
Scalar Triple Product
Vector Equation of a Line
Parametric Equations of a Line
Symmetric Equations of a Line
Vector Equation of a Plane
Scalar Equation of a Plane
Distance from a Point in Space to a Line
Distance from a Point in Space to a Plane
Distance Between Two Parallel Planes
Direction Angles and Direction Cosines:
Directions angles are the angles made between a vector and the positive x,y and z axes. Direction cosines are the cosines of those angles. Given the vector "a" (shown below):
...where:
Direction Cosines:
Direction Angles:
Unit vector:
Given a vector "a", the unit vector of "a" has a length (magnitude) of 1 in the same direction of "a".
Dot Product:
Given vectors "a" and "b", where:
...the dot product of "a" and "b" is:
Also note that if:
Vector Projection:
The vector projection of vector "b" onto a vector "a" is known as a "vector projection" and is symbolized as:
One way of thinking about vector projection is to imagine rays of light shining towards vector "a" at perpendicular angles all along its length. The shadow cast along "a" as a result of the rays of light hiting "b" is the vector projection of "b" onto "a". Mathematically we have the following expression for this vector projection:
Scalar Projection:
The "scalar projection of vector "b" onto vector "a" (sometimes called "component projection") is the signed magnitude of the vector projection. In other words it is the length of the resulting vector projection. It is symbolized as:
Mathematically, this scalar projection is defined as:
Cross Product:
The cross product of two vectors produces a vector that is perpendicular to both.. Given the following two vectors:
Additionally, the length of this resulting cross product vector can be defined as:
Scalar Triple Product:
The scalar triple product can be used to find the volume of a parallelepiped. When given three vectors, it is found by evaluating the following determinant:
Vector Equation of a Line:
Consider the following 3-D graph of a line "L":
Let P_0 and P be points on line "L" and let "r" and "r_0" be position vectors to each of those points. The vector "a" represents the vector from point P_0 to point P. Additionally, "v" is a vector that is parallel to line "L". By the triangle law for vector addition we have:
For every value of "t", we get a position vector "r" for a point on line "L". As the value of "t" is varied, the line "L" is traced out by the tip of vector "r".
Parametric Equations of a Line:
Given a line that passes through a point:
Symmetric Equations of a Line:
If we take the parametric equations for a line and solve for "t" we get:
Vector Equation of a Plane:
Consider the following graph of a plane:
...where:
Scalar Equation of a Plane:
The scalar equation for a plane is:
Distance from a Point in Space to a Line:
Consider the following graph showing a point in space and a line:
In order to determine the distance (d) from the point to the line:
Distance from a Point in Space to a Plane:
1) Determine a point on the plane by setting y and z of the plane equation to zero and getting a point such as the following:
Distance Between Two Parallel Planes: