1 · Dot Product, Angles, and Direction Cosines
Dot product, in components and geometrically:
$$ \mathbf a\cdot\mathbf b=\sum_i a_i\,b_i=|\mathbf a|\,|\mathbf b|\cos\angle(\mathbf a,\mathbf b) $$
Angle between two vectors:
$$ \cos\angle(\mathbf a,\mathbf b)=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a|\,|\mathbf b|} $$
Direction cosine by component (dot product with a basis vector). The direction cosines are the components of the unit vector \(\hat{\mathbf a}=\mathbf a/|\mathbf a|\):
$$ \cos\angle(\mathbf a,\hat{\mathbf e}_i)=\frac{\mathbf a\cdot\hat{\mathbf e}_i}{|\mathbf a|}=\frac{a_i}{|\mathbf a|} $$
Sum of squared direction cosines.
$$ \sum_i \cos^{2}\angle(\mathbf a,\hat{\mathbf e}_i)=1 $$
2 · Cross Product and the Levi-Civita Symbol
The permutation symbol \(\varepsilon_{ijk}\), totally antisymmetric (\(\varepsilon_{ijk}=-\varepsilon_{jik}=\varepsilon_{jki}\)):
$$ \varepsilon_{ijk}=\begin{cases}+1 & (i,j,k)\ \text{cyclic of }(1,2,3)\\ -1 & \text{anticyclic}\\ \;\;0 & \text{repeated index}\end{cases} $$
Cross product, in Einstein summation convention:
$$ (\mathbf a\times\mathbf b)_i=\varepsilon_{ijk}\,a_j\,b_k,\qquad |\mathbf a\times\mathbf b|=|\mathbf a|\,|\mathbf b|\sin\angle(\mathbf a,\mathbf b) $$
Right-handed orthonormal basis, cyclic relations:
$$ \hat{\mathbf e}_1\times\hat{\mathbf e}_2=\hat{\mathbf e}_3,\qquad \hat{\mathbf e}_2\times\hat{\mathbf e}_3=\hat{\mathbf e}_1,\qquad \hat{\mathbf e}_3\times\hat{\mathbf e}_1=\hat{\mathbf e}_2 $$
3 · Vector Identities
BAC minus CAB (double cross product).
$$ \mathbf a\times(\mathbf b\times\mathbf c)=\mathbf b\,(\mathbf a\cdot\mathbf c)-\mathbf c\,(\mathbf a\cdot\mathbf b) $$
The \(\varepsilon\)-\(\delta\) identity, with one index contracted:
$$ \sum_i \varepsilon_{ijk}\,\varepsilon_{ilm}=\delta_{jl}\,\delta_{km}-\delta_{jm}\,\delta_{kl} $$
Product of two \(\varepsilon\)'s as a determinant of deltas. Useful contractions: \(\sum_{ij}\varepsilon_{ijk}\varepsilon_{ijl}=2\,\delta_{kl}\), and \(\sum_{ijk}\varepsilon_{ijk}^{2}=6\).
$$ \varepsilon_{ijk}\,\varepsilon_{lmn}=\begin{vmatrix}\delta_{il}&\delta_{im}&\delta_{in}\\ \delta_{jl}&\delta_{jm}&\delta_{jn}\\ \delta_{kl}&\delta_{km}&\delta_{kn}\end{vmatrix} $$
Scalar triple product (it is a determinant). Cyclic: \(\mathbf a\cdot(\mathbf b\times\mathbf c)=\mathbf b\cdot(\mathbf c\times\mathbf a)\). Equal to zero if the three vectors are coplanar; its absolute value is the volume of the parallelepiped they span.
$$ \mathbf a\cdot(\mathbf b\times\mathbf c)=\varepsilon_{ijk}\,a_i\,b_j\,c_k=\begin{vmatrix}a_1&a_2&a_3\\ b_1&b_2&b_3\\ c_1&c_2&c_3\end{vmatrix} $$
4 · Kinematics, and the Subtlety of \(\dot r\)
Velocity, acceleration, speed:
$$ \mathbf v=\dot{\mathbf r},\qquad \mathbf a=\ddot{\mathbf r},\qquad \text{speed}=|\mathbf v|=|\dot{\mathbf r}| $$
Careful: \(\dot r=\dfrac{d}{dt}|\mathbf r|\) is the radial velocity, not \(|\mathbf v|\). The two coincide only when the motion is purely radial, i.e. when \(\angle(\mathbf r,\mathbf v)=0\).
$$ \dot r=\frac{d}{dt}|\mathbf r|=\frac{\mathbf r\cdot\mathbf v}{|\mathbf r|}=|\mathbf v|\cos\angle(\mathbf r,\mathbf v) $$
5 · Arc Length and the Chain Rule
Arc length measured from \(t_0\):
$$ s(t)=\int_{t_0}^{t}|\dot{\mathbf r}(u)|\,du,\qquad \frac{ds}{dt}=|\dot{\mathbf r}(t)| $$
Passing from \(t\) to \(s\). The factor that is almost always forgotten.
$$ \frac{d}{ds}=\frac{1}{|\dot{\mathbf r}|}\,\frac{d}{dt} $$
6 · Curvature and Torsion
Curvature from \(\mathbf r(t)\) — the "I'm given the function" case.
$$ \kappa=\frac{|\dot{\mathbf r}\times\ddot{\mathbf r}|}{|\dot{\mathbf r}|^{3}}=\frac{|\mathbf v\times\mathbf a|}{|\mathbf v|^{3}} $$
Curvature, defined in the arc-length parameter:
$$ \kappa=\left|\frac{d\hat{\mathbf t}}{ds}\right| $$
Torsion, definition and solved via the dot product:
$$ \frac{d\hat{\mathbf b}}{ds}=-\tau\,\hat{\mathbf n}\qquad\Longrightarrow\qquad \tau=-\frac{d\hat{\mathbf b}}{ds}\cdot\hat{\mathbf n} $$
Torsion from \(\mathbf r(t)\), via the scalar triple product.
$$ \tau=\frac{\dot{\mathbf r}\cdot(\ddot{\mathbf r}\times\dddot{\mathbf r})}{|\dot{\mathbf r}\times\ddot{\mathbf r}|^{2}} $$
Radius of curvature. In the arc-length parameter, \(\kappa=|\ddot{\mathbf r}|\) and \(\dot{\mathbf r}\cdot(\ddot{\mathbf r}\times\dddot{\mathbf r})=\kappa^{2}\,\tau\).
$$ \rho=\frac{1}{\kappa} $$
7 · The Moving Trihedron
Tangent (unit vector of the velocity):
$$ \hat{\mathbf t}=\frac{\dot{\mathbf r}}{|\dot{\mathbf r}|} $$
Principal normal. In a general parameter it is often more convenient to use \(\hat{\mathbf n}=\hat{\mathbf b}\times\hat{\mathbf t}\), which avoids differentiating \(\hat{\mathbf t}\).
$$ \hat{\mathbf n}=\frac{1}{\kappa}\,\frac{d\hat{\mathbf t}}{ds} $$
Binormal. Computed directly from \(\mathbf r(t)\): \(\hat{\mathbf b}=\dfrac{\dot{\mathbf r}\times\ddot{\mathbf r}}{|\dot{\mathbf r}\times\ddot{\mathbf r}|}\). The osculating plane is perpendicular to \(\hat{\mathbf b}\), the normal plane to \(\hat{\mathbf t}\), and the rectifying plane to \(\hat{\mathbf n}\).
$$ \hat{\mathbf b}=\hat{\mathbf t}\times\hat{\mathbf n} $$
8 · The Frenet-Serret Formulas
The three formulas, in the arc-length parameter.
$$ \frac{d\hat{\mathbf t}}{ds}=\kappa\,\hat{\mathbf n},\qquad \frac{d\hat{\mathbf n}}{ds}=-\kappa\,\hat{\mathbf t}+\tau\,\hat{\mathbf b},\qquad \frac{d\hat{\mathbf b}}{ds}=-\tau\,\hat{\mathbf n} $$
Matrix form, antisymmetric (an instantaneous rotation):
$$ \frac{d}{ds}\begin{pmatrix}\hat{\mathbf t}\\ \hat{\mathbf n}\\ \hat{\mathbf b}\end{pmatrix}=\begin{pmatrix}0&\kappa&0\\ -\kappa&0&\tau\\ 0&-\tau&0\end{pmatrix}\begin{pmatrix}\hat{\mathbf t}\\ \hat{\mathbf n}\\ \hat{\mathbf b}\end{pmatrix} $$
If the trihedron is known as a function of \(t\) rather than \(s\), multiply each formula by \(ds/dt=|\dot{\mathbf r}|\):
$$ \frac{d\hat{\mathbf t}}{dt}=\kappa\,|\dot{\mathbf r}|\,\hat{\mathbf n},\qquad \frac{d\hat{\mathbf b}}{dt}=-\tau\,|\dot{\mathbf r}|\,\hat{\mathbf n} $$
9 · Special Cases
Graph \(y=f(x)\) — curvature.
$$ \kappa=\frac{|f''|}{\big(1+f'^{\,2}\big)^{3/2}} $$
Circle of radius \(R\):
$$ \kappa=\frac{1}{R} $$
Test for a planar curve (equivalent conditions):
$$ \tau\equiv 0 \;\Longleftrightarrow\; \dot{\mathbf r}\cdot(\ddot{\mathbf r}\times\dddot{\mathbf r})=0 \;\Longleftrightarrow\; \hat{\mathbf b}=\text{constant} $$