a phasor
Tracking a sine wave that is both swinging in size and shifting in time gets messy with wiggly curves. A phasor is a clever shortcut: freeze the rotating wave into a single arrow. Its length is the amplitude and the direction it points is the phase. Picture the sine as the shadow of a spinning arrow; the phasor is that arrow caught at one instant.
Since every sine in a linear circuit shares the same frequency, you can ignore the common spinning and just record each signal's size and phase as a stationary arrow (a vector, or equivalently a complex number). Adding two sine signals becomes adding two arrows tip-to-tail; a 90-degree phase shift becomes a quarter-turn of the arrow. The painful calculus of dV/dt turns into simple arrow arithmetic.
Phasors turn impedance, filters, and AC analysis into geometry and algebra instead of differential equations, the everyday working tool of AC circuit design. Caveat: the trick works only for steady sine waves of one frequency in a linear circuit. It says nothing about switch-on transients, square waves, or non-linear parts, where you must go back to the full time-domain picture.
Adding a 3 V signal to a 4 V signal that lags it by 90 degrees does not give 7 V. Drawn as two arrows at right angles, the sum is the square root of (3^2 + 4^2) = 5 V.
Out-of-phase signals add like arrows, not numbers.
Phasors only work for steady single-frequency sines in linear circuits; they cannot represent transients, square waves, or non-linear behaviour.