Why is the $90$ degree clockwise rotation matrix not representative of the locations of $\hat\imath$ and $\hat\jmath$?

Question

I was watching a 3Blue1Brown video on inverse matrices and he showed that a $90$ degree counterclockwise rotation has the matrix $\begin{bmatrix}0 &-1 \\ 1 & 0\end{bmatrix}$ which makes sense to me since it represents where $\hat\imath$ and $\hat\jmath$ land. But then he shows the inverse matrix that represents a clockwise rotation, with the matrix $\begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}$ which does not correspond to $\hat\imath$ and $\hat\jmath$. Why is this?

Answer 1

Let us label the matrices by

$$R=\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad R^{-1} =\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \newcommand{\i}{\hat\imath} \newcommand{\j}{\hat\jmath}$$

Note that the first column responds to where $\hat\imath$ goes, and the second where $\j$ goes, after applying the matrix to $\mathbb{R}^2$ in its normal state.

So:

• $R$ sends $\i$ to $(0,1)$, matching a $90^\circ$ counterclockwise rotation
• $R$ sends $\j$ to $(-1,0)$, likewise matching
• $R^{-1}$ sends $\i$ to $(0,-1)$, matching a $90^\circ$ clockwise rotation
• $R^{-1}$ sends $\j$ to $(1,0)$, likewise matching

You can play with this in this Desmos demo, which essentially rotates a given vector $(a,b)$ by a more general rotation matrix. Some notes:

• It assumes a rotation by $T\pi$ radians counterclockwise. $90^\circ$ clockwise corresponds to $3\pi/2$ radians counterclockwise as a result.
• The red vector displayed is the original, and the black the result.