Mysterious vector operations

Question

My physics professor just gave this definition for vector multiplication and division that I can't corroborate anywhere:

$$ \begin{align} \mathbf{A} &= 9 \angle 37°\\ \mathbf{B} &= 3 \angle 20°\\ \mathbf{A} * \mathbf{B} &= |\mathbf{A}||\mathbf{B}|\angle(\theta_A + \theta_B) = 27\angle57°\\ \frac{\mathbf{A}}{\mathbf{B}} &= \frac{|\mathbf{A}|}{|\mathbf{B}|}\angle(\theta_A - \theta_B) = 3\angle17° \end{align} $$

My physics and multivariable calculus texts only mention the dot and cross product and explicitly don't define division. After several google and stack exchange searches turned up nothing, I'm here.

Also, if it helps identify the origin of these operations, the professor is an engineer.

Answer 1

These are phasor notations.

In $A = M\angle{ (\omega t+ \theta)}$, $M$ is the amplitude and $\theta$ is the phase shift.

This can be equivalently written as $Me^{j(\omega t+\theta)}$, if angular frequency $\omega$ is considered.

So, let $A = 9\angle37^o = 9e^{j37^o}$

$B = 3\angle20^o = 3e^{j20^o}$

So,

$$A\cdot B = 9e^{j37^o}3e^{j20^o} = 27 e^{j 57^o} = 27\angle57^o$$

Also,

$$\frac{A}{B} = \frac{9e^{j37^o}}{3e^{j20^o}} = 3e^{j37^o -j20^o} = 3e^{j17^o} =3\angle17^o$$