Explain the representation of $AC$ current and voltage by rotating vectors.

(N/A) The $AC$ current through a resistor is in phase with the voltage. However,this is not the case for an inductor,a capacitor,or a combination of these circuit elements.
To show the phase relationship between voltage and current in an $AC$ circuit,it is convenient to analyze the circuit using the concept of phasors.
$A$ phasor is a vector that rotates about the origin with an angular speed $\omega$,as shown in the figure.
The vertical components of the phasors $V$ and $I$ are $V_{m} \sin \omega t$ and $I_{m} \sin \omega t$,where $V_{m}$ and $I_{m}$ represent the peak values of the oscillating quantities.
Figure $(a)$ shows the voltage and current phasors and their relationship at time $t_{1}$ for a resistor connected to an $AC$ source.
The projections of the $V$ and $I$ phasors on the vertical axis,$V_{m} \sin \omega t_{1}$ and $I_{m} \sin \omega t_{1}$,represent the instantaneous values of voltage and current.
Figure $(b)$ shows the corresponding sinusoidal waveforms generated by these rotating vectors with frequency $\omega$.
For a resistor,the phasors $V$ and $I$ are in the same direction,which means the phase angle between the voltage and the current is zero.