How come multiplying this unit vector by another is a rotation? I was reading a book about rotations today and in the introduction there are two complex vectors $\langle 1,1 \rangle = [1,1]$ and $\langle 0,1\rangle = [0,1]$. I initially thought the columns of one didn't match the rows of the other because it doesn't show that computation. The result of multiplying $\langle 1,1 \rangle$ by $\langle 0,1\rangle$ rotates $\langle 1,1 \rangle$ by the other vector's argument $\pi/2$ to $\langle -1,1\rangle$.
I've checked the cross product of the vectors and it doesn't look like it exists, because there are no entries to compute the determinant of the remaining column with after removing the first or second entry.
More specifically I thought if these are vectors that are row matrices with a row for each dimension they would be the row vectors $[1 , 1]$ and $[0,1]$. I was sure they can be multiplied but the columns and rows do not match so what do I do different?
 A: One can represent complex numbers $a+bi$ as vectors in the plane $[a,b]$.  It works nicely for addition because complex addition is componentwise as is our usual addition of vectors.  Complex multiplication is completely different from any vector product we are used to.  We can just define the multiplication of two of these vectors as $[a,b]\cdot[c,d]=[ac-bd,ad+bc]$  This matches the usual multiplication of complex numbers.  Normally in two dimensions there is no vector multiplication that results in a vector.
A: You can treat the complex numbers as the vector space $\mathbb{R}^2$ (i.e. the Cartesian plane) with the complex number $a + ib$ being equated with the vector $(a, b)$. However, $\mathbb{R}^2$ does not come with its own multiplication operation - like any vector space, you can add vectors to each other, and you can multiply vectors by real numbers, but $(a, b) \times (c, d)$ has no intrinsic definition.
There are, in fact, multiple options for a multiplication operation on $\mathbb{R}^2$ depending on what properties you want to have - for example, $(a, b) \times (c, d) = (ac, bd)$ is a valid choice under some circumstances (and makes sense if you map the vectors to diagonal 2x2 matrices).
In our case, though, the aim is to make complex arithmetic a smooth extension of real arithmetic, and in particular we know that $i^2 = -1$ so in vector form it means we're stuck with having $(0, 1) \times (0, 1) = (-1, 0)$. As it turns out, the way to do this is to define multiplication as $(a, b) \times (c, d) = (ac - bd, ad + bc)$. It doesn't relate to matrix multiplication at all.
Except ...
We can also write vectors in $\mathbb{C} = \mathbb{R}^2$ in polar coordinates, i.e. in terms of their distance from the origin and counterclockwise from the positive $x$-axis, which gives us $x + iy = (x, y) = (r \cos \theta, r \sin \theta) = r \cos \theta + i \sin \theta = r\ \mathrm{cis}\ \theta$. In this notation, you can show that $r\ \mathrm{cis}\ \theta \times s\ \mathrm{cis}\ \psi = rs\ \mathrm{cis}\ (\theta + \psi)$ - i.e. complex multiplication multiplies the vector lengths and adds the angles.
And, as it happens, there is a way to do vector rotation using matrices, and we can steal that to give a matrix representation for our complex numbers. If we write:
$$x + iy \sim \begin{bmatrix} x & y \\
-y & x \end{bmatrix}$$
then it turns out that this 2x2 matrix representation gives us both complex addition and multiplication using the standard matrix operations.
In all of these cases, though, it can be useful to write the "vector that's actually a complex number" differently to "just a normal real-valued 2-dimensional vector" so that it's clear that we have this extra multiplication operation available to us, which is why we may use the square or angle brackets.
A: You can identify the vector $\langle a,b\rangle$ (the position vector of a point $(a,b)$ in the plane) as the complex number $a+bi$ in the Argand plane. Multiplying by $\langle\cos\theta,\sin\theta\rangle$, ie, $\cos\theta+i\sin\theta$ then gives a counter-clockwise rotation by angle $\theta$. As such, multiplying by $\langle 0,1\rangle=\langle\cos\frac\pi 2,\sin\frac\pi 2\rangle$ gives a counter-clockwise rotation by $\dfrac\pi 2$ radians, ie, $90^\circ$
The operations are defined as how it is defined for $\Bbb C$ when it is constructed from $\Bbb R^2$ by,
$$\langle a,b\rangle+\langle c,d\rangle=\langle a+c,b+d\rangle\\ \lambda\langle a,b\rangle=\langle\lambda a,\lambda b\rangle\quad\text{for scalar }\lambda\in\Bbb R\\\langle a,b\rangle\cdot\langle c,d\rangle=\langle ac-bd,ad+bc\rangle$$
