In an inertial frame of reference (let's say a car moving with certain constant velocity in which I am sitting) If I observe the motion of another car through my frame of reference i.e.car, will I be at rest with my frame of reference or in motion?
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You're always at rest with respect to your own frame. It doesn't matter what else is going on.
If we model you as a "disembodied spatial tangent vector" (i.e. a point particle with a preferred spatial direction to represent your line of sight), then you'll need to spin in place to keep an eye on the other guy, but that doesn't count as motion.
If we think of you as three dimensional, then parts of your body will have to move, but there's no need for your center of mass to move.
If I observe the motion of another car through my frame of reference i.e.car, will I be at rest with my frame of reference or in motion?
You will always be "at rest" in your own frame of reference. You will not be at rest with respect to the frame of reference of the person in the other car. Nor will the person in the other car be "at rest" with respect to your frame of reference.
The terms "at rest" and "moving" only have meaning with respect to a particular reference frame. An object cannot be both at rest and moving in the same reference frame. The exception is the speed of light in a vacuum which is the same with respect to any inertial frame of reference.
Hope this helps.