[Physics] What exactly is a phasor

Question

What exactly is a phasor? I was reading about alternating current when I came across the following definition:

A phasor is a vector which rotates about the origin with an angular speed(suppose $\omega$).

Then the book mentions the following statement: Though voltage and current in an AC circuit are represented by phasors-rotating vectors, they are not vectors themselves.

Aren't the 2 statements contradictory?

In my knowledge, a vector quantity is one which follows the law of vector addition(correct me if I'm wrong).

The book even obtains the impedence of an LCR circuit by using phasors and adding them just like vectors. So, what exactly is the difference between the two?

Answer 1

Think of a combination of the complex plane and ordinary vectors.

A phasor is a complex number,  representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant.

Image and text from Phasors Wikipedia

Assume you have a network composed of multiple sinusoids (waves). They all have the same frequency, but with differing amplitudes and phases. The only difference in their analytic representations is the complex amplitude (phasor). A linear combination of such functions can be factored into the product of a linear combination of phasors (known as phasor arithmetic) and the time/frequency dependent factor that they all have in common.

When function ${\displaystyle \scriptstyle A\cdot e^{i(\omega t+\theta )}}$ is depicted in the complex plane, the vector formed by its imaginary and real parts rotates around the origin. Its magnitude is $A$, and it completes one cycle every $2π/ω$ seconds. $θ$ is the angle it forms with the real axis at $t = n•2π/ω$, for integer values of n.