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The length is multiplied by the absolute value of the scalar $|c|$. Specifically, $0 < |c| < 1$ shortens the vector, $|c| > 1$ stretches it, and $c=0$ yields the zero vector.

Conditions: Scalar multiplication operation; Considering magnitude/length changes

The length is multiplied by the absolute value of the scalar $|c|$. Specifically, $0 < |c| < 1$ shortens the vector, $|c| > 1$ stretches it, and $c=0$ yields the zero vector.

Conditions: Scalar multiplication operation; Considering magnitude/length changes

Multiplying $g(t)$ by $e^{-2\pi i ft}$ rotates the signal's amplitude around the complex plane at a rate determined by $f$. When averaged over time, contributions from frequencies matching $f$ accumulate constructively (moving away from the origin), while mismatched frequencies tend to cancel out due to symmetric distribution around the circle.

Conditions: $g(t)$ is a real-valued time-domain signal.; $f$ represents cycles per unit time.; The averaging is performed over a finite or infinite time window.

Multiplying $g(t)$ by $e^{-2\pi i ft}$ rotates the signal's amplitude around the complex plane at a rate determined by $f$. When averaged over time, contributions from frequencies matching $f$ accumulate constructively (moving away from the origin), while mismatched frequencies tend to cancel out due to symmetric distribution around the circle.

Conditions: $g(t)$ is a real-valued time-domain signal.; $f$ represents cycles per unit time.; The averaging is performed over a finite or infinite time window.