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## Homework Statement

Consider the unit vector, [tex]\hat{v}(t)[/tex], expressed in instantaneous form:

[tex]\hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y}[/tex] (#0)

The Vector will rotate counterclockwise in the x-y plane with angular velocity [tex]\omega[/tex].

Since both components are sinusoidally time varying, and since there is a 90degrees phase shift between the components, we can express this vector as:

[tex]\hat{v} = (1 + j0)\hat{x} + (0 + j)\hat{y}[/tex] (#1)

**2. Relevant Question**

How was the equation above (equation #1) defined?

**3. Thought Process**

By Euler's Identity, phasors can be written as,

[tex]Ae^{j\phi} = {Acos(\phi), Asin(\phi)}[/tex] (real, and imaginary parts respectively)

Can we relate the identity above somehow to change equation (#0) into something like [tex]\hat{v}(t) = cos(\omega t)\hat{x} + jsin(\omega t)\hat{y}[/tex]

And if we take the function of "t" out from equation (#0), why wouldn't equation (#1) become:

[tex]\hat{v} = cos(\omega)\hat{x} + jsin(\omega )\hat{y}[/tex]

Thanks,

JL

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