To find the phasor notation for the given time-harmonic vector:

$V(t) = 3\cos(\omega t) \hat{x} + 4\sin(\omega t) \hat{y} + \cos(\omega t + \frac{\pi}{2}) \hat{z}$

Step-by-Step Solution:

1. Express the components in terms of cosines:

   Recall the following trigonometric identity for sine in terms of cosine: $\sin(\omega t) = \cos\left(\omega t - \frac{\pi}{2}\right)$ So, we can rewrite the components of the vector in cosine form:

   • For the $\hat{x}$-component, $3 \cos(\omega t)$ is already in cosine form, so it remains as is.

   • For the $\hat{y}$-component, $4 \sin(\omega t)$ becomes: $4 \sin(\omega t) = 4 \cos\left(\omega t - \frac{\pi}{2}\right)$

   • For the $\hat{z}$-component, $\cos(\omega t + \frac{\pi}{2})$ can be rewritten using the phase-shift identity: $\cos(\omega t + \frac{\pi}{2}) = -\sin(\omega t) = -\cos\left(\omega t - \frac{\pi}{2}\right)$

2. Represent the components in phasor form:

   The phasor notation involves converting the time-dependent cosines into complex exponentials. A general cosine term $A \cos(\omega t + \phi)$ can be represented as a phasor: $A \angle \phi$ Where $A$ is the amplitude and $\phi$ is the phase shift.

   Now, we convert each term:

   • The $\hat{x}$-component: $3 \cos(\omega t)$ corresponds to a phasor $3 \angle 0^\circ$.

   • The $\hat{y}$-component: $4 \cos(\omega t - \frac{\pi}{2})$ corresponds to a phasor $4 \angle -90^\circ$.

   • The $\hat{z}$-component: $-\cos(\omega t - \frac{\pi}{2})$ corresponds to a phasor $-1 \angle -90^\circ$, which is equivalent to a phasor with amplitude 1 and phase $-90^\circ$.

Final Phasor Representation:

Now we can write the phasor notation for the vector $V(t)$ as:

$V = 3 \angle 0^\circ \hat{x} + 4 \angle -90^\circ \hat{y} - 1 \angle -90^\circ \hat{z}$

This is the phasor form of the given time-harmonic vector.

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Would you like further explanation or assistance with any other problem?

Related Questions:

1. How do you convert a sine function to its phasor form?
2. How do you express a vector in terms of its real and imaginary parts in phasor notation?
3. What is the meaning of a phase shift in phasor notation?
4. How do you add multiple phasors together?
5. What is the difference between complex exponential representation and phasor representation?

Tip:

When working with phasors, remember that the phasor represents the amplitude and phase of a cosine function. By using Euler’s formula, you can relate a cosine term to its complex exponential equivalent.