This page covers the mathematical notation used in a number of the posts on this blog. As force of habit I can’t help but use “we” in mathematical writing – I hope it doesn’t come across as too pretentious!

Linear Algebra

Components and Unit Vectors

We write $I$ for the identity matrix when its dimensions can be inferred. When discussing a vector $v \in \mathbb{R}^n$ we denote its $i$-th component by $v[i]$. When discussing $\mathbb{R}^n$ we define $e_i$ via:

$$ e_i[j] := \begin{cases} 1 & i = j \\ 0 & \textrm{o.w.}\end{cases} $$

for $i \in {1, \ldots, n}$. This is equivalent to saying $e_i$ is the $i$-th column of the identity matrix. With this choice we can write:

$$ v[j] = e_j^T v $$

For $\mathbb{R}^3$ we write the standard unit vectors as: $$ e_x := e_1, \qquad e_y := e_2, \qquad e_z := e_3 $$

For $v \in \mathbb{R}^3$ we write its components as: $$ v_x := e_x^T v = v[1], \qquad v_y := e_y^T v = v[2], \qquad v_z := e_z^T v = v[3]. $$

Geometry

Cross product matrix

For a vector $u \in \mathbb{R}^3$ we’ll write $$[u]_\times = \begin{bmatrix} 0 & -u_z & u_y \\ u_z & 0 & -u_x \\ -u_y & u_x & 0\end{bmatrix}$$ to mean the $3\times 3$ matrix such that $[u]_\times v = u \times v$ for a all $v \in \mathbb{R}^3$.

Reference Frames

A reference frame will be a point in space and an associated set of right-handed axes. We denote reference frames with capital letters.

Rotations

• For a vector $v$ in three-dimensional space we write $v^A$ to denote $v$ resolved in the axes of the reference frame A.

• The symbol $C_B^A$ denotes the rotation matrix rotating vectors expressed in the coordinates of the $B$-frame to be represented in the coordinates of the $A$-frame, i.e. $v^A = C_B^A v^B$ for all $v$.

These matrices follow a simple rule: $$ C_B^C C_A^B = C_A^C $$

Positions

• The symbol $p_{AB}$ denotes the vector connecting the origin of the $A$ frame to the origin of the $B$ frame. With these definitions we some simple rules:

$$ p_{AB}^D + p_{BC}^D = p_{AC}^D $$

Time-varying Reference Frames

If two reference frames $A$ and $B$ vary with respect to time, we define:

$$ v_{AB}^A(t) := \frac{d}{dt} p_{AB}^A(t), \qquad a_{AB}^A(t) := \left(\frac{d}{dt}\right)^2 p_{AB}^A(t) $$

When resolving velocity or acceleration in another frame we write: $$ v_{AB}^C(t) := C_A^C v_{AB}^A(t), \qquad a_{AB}^C := C_A^C a_{AB}^A(t). $$

Additionally, we define $\omega_{AB}^B(t)$ to be the angular rate such that: $$ \frac{d}{dt} C_B^A(t) = C_B^A(t)[\omega_{AB}^B(t)]_\times $$