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: $$ \vec{a} = \langle a_1, a_2, a_3 \rangle = \langle |a|\cos \alpha , |a|\cos \beta, |a| \cos \Upsilon \rangle$$ ...the direction cosines and direction angles can be found through the following:

## Direction Cosines:

$$ \cos\alpha = \frac{a_1}{|a|} ,\; \cos \beta = \frac{a_2}{|a|}, \; \cos \Upsilon = \frac{a_3}{|a|} $$

## Direction Angles:

$$ \alpha = cos^{-1} \Big( \frac{a_1}{|a|} \Big), \; \beta = cos^{-1} \Big( \frac{a_2}{|a|} \Big) , \; \Upsilon = cos^{-1} \Big( \frac{a_3}{|a|} \Big)$$

### Unit vector:

Given a vector "**a**", the unit vector of "**a**" has a length (magnitude) of 1 in the same direction of "**a**".
$$ \hat{a} = unit \; vector \; in \; direction \; of \; \vec{a} $$
$$ \;\; = \frac{\vec{a}}{|\vec{a}|} \;\;\;\;\;, where \; |\vec{a}| = magnitude \; of \; \vec{a} $$
$$ \hat{a} = \frac{\vec{a}}{\sqrt{(a_1)^2+(a_2)^2+(a_3)^2}} $$

### Dot Product:

Given vectors "**a**" and "**b**", where:
$$ \vec{a} = \langle a_1, a_2, a_3 \rangle $$
$$ \vec{b} = \langle b_1, b_2, b_3 \rangle $$

...the dot product of "**a**" and "**b**" is:
$$ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 $$
or:
$$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta $$
...where:
$$ |\vec{a}| \;and\; |\vec{b}| = magnitudes \; of \vec{a} \;and\; \vec{b}$$
...and:
$$ \theta = angle \; between \; \vec{a} \;and\; \vec{b} \; when \; placed \; tail \; to \; tail. $$
note: The result of the dot product is a scalar (not a vector).

Also note that if: $$ \vec{a} \cdot \vec{b} = 0 $$ then the vectors are orthogonal (90 degrees apart).

### Vector Projection:

The vector projection of vector "**b**" onto a vector "**a**" is known as a "vector projection" and is symbolized as:
$$ proj_{\vec{a}}\vec{b} $$
Graphically it looks like the following:

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:
$$ proj_{\vec{a}}\vec{b} = \Big( \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \Big) \vec{a} $$
...where:
$$ \qquad |\vec{a}|^2 = \vec{a} \cdot \vec{a} $$
Note that the result of a vector projection is a vector quantity.

### 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:
$$ comp_{\vec{a}}\vec{b} $$
(shown below)

Mathematically, this scalar projection is defined as: $$ comp_{\vec{a}}\vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|} $$ Note that the result of a scalar projection is a scalar quantity.

### Cross Product:

The cross product of two vectors produces a vector that is perpendicular to both.. Given the following two vectors:
$$ \vec{a} = \langle a_1, a_2, a_3 \rangle $$
$$ \vec{b} = \langle b_1, b_2, b_3 \rangle $$
The cross product of those two vectors can be found by evaluating the following determinant:
$$
\vec{a} \times \vec{b} =
\begin{vmatrix}
\hat{i}&\hat{j}&\hat{k}\\
a_1&a_2&a_3\\
b_1&b_2&b_3\\
\end{vmatrix}
$$
$$
\qquad \;= \begin{vmatrix}
a_2&a_3\\
b_2&b_3\\
\end{vmatrix}
\hat{i}
\;\; -
\begin{vmatrix}
a_1&a_3\\
b_1&b_3\\
\end{vmatrix}
\hat{j}
\;\; +
\begin{vmatrix}
a_1&a_2\\
b_1&b_2\\
\end{vmatrix}
\hat{k}
$$
$$
\qquad \; = (a_2b_3-a_3b_2)\hat{i} - (a_1b_3-a_3b_1)\hat{j} + (a_1b_2-a_2b_1)\hat{k}
$$
Also, if:
$$ \vec{a} \times \vec{b} = 0 $$
...then the two vectors are parallel.

Additionally, the length of this resulting cross product vector can be defined as:
$$ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta$$
Lastly, a vector crossed by itself results in zero:
$$ \vec{a} \times \vec{a} = 0 $$

### 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: $$ \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1&a_2&a_3\\ b_1&b_2&b_3\\ c_1&c_2&c_3\\ \end{vmatrix} $$ Additionally, note that: $$ \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{c} \cdot (\vec{a} \times \vec{b}) $$

### 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:
$$ \vec{r} = \vec{r}_0 + \vec{a} $$
However, since vectors "**a**" and "**v**" are parallel, there must exist some scalar value (t) such that:
$$ \vec{a} = t\vec{v} $$
Therefore we have the following vector equation for a line:

$$ \vec{r} = \vec{r}_0 + t\vec{v} = vector \; equation \; for \; line \; "L"$$

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: $$ P_0(x_0,y_0,z_0) $$ and is parallel to the vector: $$ \vec{v} = \langle a,b,c \rangle $$ ...where a,b and c are the "direction numbers" of the line, the parametric equations for the line is: $$ x = x_0+at $$ $$ y = y_0+bt $$ $$ z = z_0+ct $$ Each value of t gives a single point on the line.

### Symmetric Equations of a Line:

If we take the parametric equations for a line and solve for "t" we get: $$ x = x_0+at \qquad, y = y_0+bt \qquad, z = z_0+ct $$ $$ t = \frac{x-x_0}{a} \qquad , t = \frac{y-y_0}{b} \qquad, t = \frac{z-z_0}{c} $$ This gives us the following expression for t: $$ t = \frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c} $$ ...where a,b and c are the "direction numbers" of the line.

### Vector Equation of a Plane:

Consider the following graph of a plane:

...where: $$ P,P_0 = points \; in \; plane $$ $$ \vec{r}, \vec{r}_0 = position \; vectors \; of \; pts. \; P,P_0 $$ $$ \vec{r} - \vec{r}_0 = \langle P(x,y,z)-P_0(x_0,y_0,z_0) \rangle $$ $$ \vec{n} = \langle a,b,c \rangle = vector \; orthogonal \; to \; plane $$ $$ \vec{r} = \langle x,y,z \rangle $ = vector \; to \; point \; on \; plane$$ $$ \vec{r}_0 = \langle x_0, y_0, z_0 \rangle = vector \; to \; point \; on \; plane $$ The vector equation of a plane is: $$ \vec{n} \cdot (\vec{r}-\vec{r}_0) = 0 $$ ...which can also be written as: $$ \vec{n} \cdot \vec{r} = \vec{n} \cdot \vec{r}_0 $$

### Scalar Equation of a Plane:

The scalar equation for a plane is: $$ a(x-x_0)+b(y-y_0)+c(x-x_0)=0 $$ If we collect like terms in the above equation we get the linear equation of the plane: $$ ax+by+cz+d=0 $$

### 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: $$1) \; Get \; the \; direction \; vector \; of \; the \; line \; (\vec{V}_L) $$ $$2) \; Find \; a \; coordinate \; of \; a \; point \; on \; the \; line $$ $$ \quad (Set \; "t" \; equal \; to \; number \; in \; parametric \; eqns.)$$ $$ 3) \; Get \; the \; direction \; vector \; from \; pt. \; in \; space \; to \; pt. \; on \; line. (\vec{V}_d)$$ The distance is found through the following: $$ d = \frac{|\vec{V}_d \times \vec{V}_L|}{|\vec{V}_L|} $$ Recall that: $$ \qquad \quad|\vec{V}_d \times \vec{V}_L| = |\vec{V}_d | |\vec{V}_L| \sin(\theta) $$ $$ Therefore: $$

$$ d = |\vec{V}_d| \sin(\theta) $$

### 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: $$ (number,0,0) $$ 2) Determine the vector from this point on the plane (P0) to the point in space (P1). $$ This \; vector\; = \vec{b} $$ 3) Get the normal vector of the plane: $$ \qquad \vec{n} = \langle a,b,c \rangle $$ The distance from the pt. in space to the plane is now found by: $$ d = comp_{\vec{n}}\vec{b} = \frac{\vec{n} \cdot \vec{b}}{|\vec{n}|} $$

### Distance Between Two Parallel Planes:

$$ D = \frac{ax_1+by_1+cz_1+d}{\sqrt{a^2+b^2+c^2}} $$ ...where: $$ (x_1, y_1, z_1) \; is \; pt. \; on \; plane $$ $$ a,b,c = coefficients \; of \; the \; plane \; that \; doesn't \; contain \; the \; above \; pt. $$