Reconsider the direction cosine matrix notation

[moorepants]

We (the course instructors) have discussed this notation several times over the years:

If you think of $\hat{a}_x$ representing a column vector $[1, 0, 0]^T_A$. Then the above notation implies that there is a $9 \times 1 = 3 \times 3 \cdot 9 \times 1$ matrix multiplication happening.

It is true you can write such an equation:

$\hat{a}_x = cos(q) \hat{b}_x + sin(q) \hat{b}_y$

in this notation where $\hat{u}$ is a unit vector. So, it seems natural to put the coefficients of the $\hat{b}$ unit vectors into a matrix even though they are not scalars.

In Kane and Levinson's spacecraft book they use this:

which does retain the correct dimensionality if the unit vectors are considers column vectors.

Note that this video by David Levinson: https://nescacademy.nasa.gov/video/6a798400b885492fb854449ca1fa72061d

uses the notation that I have.