If any two vectors are identical or parallel, STP is 0.

Volume of Parallelepiped: $|[\vec{a} \vec{b} \vec{c}]|$

Volume of Tetrahedron: $\frac{1}{6}|[\vec{a} \vec{b} \vec{c}]|$

Coplanarity: If $\vec{a}, \vec{b}, \vec{c}$ are coplanar, then $[\vec{a} \vec{b} \vec{c}] = 0$.

Definition: $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$ (BAC - CAB rule)
Note: $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a}$

Distance Formula: Between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.
Section Formula:
- Internal: $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n}\right)$
- External: $\left(\frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}, \frac{mz_2-nz_1}{m-n}\right)$
Centroid of Triangle: $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$
Centroid of Tetrahedron: $\left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4}\right)$

Direction Cosines (DC's): $\ell = \cos\alpha, m = \cos\beta, n = \cos\gamma$. $\ell^2+m^2+n^2=1$.
Direction Ratios (DR's): Any set $a, b, c$ proportional to DC's. $\frac{\ell}{a} = \frac{m}{b} = \frac{n}{c} = \frac{1}{\sqrt{a^2+b^2+c^2}}$.
DC's of line joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $\frac{x_2-x_1}{d}, \frac{y_2-y_1}{d}, \frac{z_2-z_1}{d}$ where $d$ is the distance.
Angle between two lines: $\cos\theta = \ell_1\ell_2 + m_1m_2 + n_1n_2$ (using DC's). Or $\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ (using DR's).

Equation of line through $(x_1, y_1, z_1)$ with DR's $a, b, c$:
- Cartesian: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$
- Vector: $\vec{r} = \vec{a} + \lambda\vec{b}$, where $\vec{a}$ is position vector of point, $\vec{b}$ is direction vector.
Equation of line through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$:
- Cartesian: $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$
- Vector: $\vec{r} = \vec{a} + \lambda(\vec{b}-\vec{a})$
Angle between two lines: $\cos\theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}$ (vector form).
Shortest Distance between Skew Lines:
- Lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$
- $d = \left|\frac{(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})}{|\vec{b_1} \times \vec{b_2}|}\right|$
Shortest Distance between Parallel Lines:
- Lines $\vec{r} = \vec{a_1} + \lambda\vec{b}$ and $\vec{r} = \vec{a_2} + \mu\vec{b}$
- $d = \left|\frac{(\vec{a_2} - \vec{a_1}) \times \vec{b}}{|\vec{b}|}\right|$

Equation of Plane:
- Normal Form: $\vec{r} \cdot \hat{n} = d$, where $\hat{n}$ is unit normal and $d$ is distance from origin.
- Cartesian Normal Form: $\ell x + my + nz = d$
- General Form: $Ax+By+Cz+D=0$. Normal vector $\vec{n} = A\hat{i} + B\hat{j} +