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I have continuously searched for this question to get an understanding of what is going on, but I seem to only find problems where the radius is given or there are only 2 coordinates.

Parametrize the circle of radius R centered at (7,1,-2) on the plane x+2y+3z=3.

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    What ways do you have to get started? Hint: $R$ isn't really a variable. Try it with a number first. – 2017-01-26

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You need to find a vector normal to the plane. One such is obtained using the coefficients of x,y,z so that $n=(1,2,3)$ is normal to the plane. Now you also need a pair of unit vectors each normal to $n$ and also normal to each other. This can be done using say $(2,-1,0)$ for one of them and then cross product for the third, and to make them unit vectors divide each by its length. Call these $u,v.$

Then a parametrization would be $(x,y,z)=(7,1,-2)+R(\cos t)u +R (\sin t) v.$