Parametrization of a circle (extra credit on calculus final) This was an extra credit question on my Calculus final. Parametrize the circle lying in the plane with normal vector $(1, 1, -2)$ with center at $$\Big(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\Big)$$ and passing through the origin. I figured out that looking down the $z$ axis the "circle" would resemble a ellipse, while along the $x$ or $y$ axis it would look like a line with the values bobbing back and forth periodically but I wasn't able to convince myself of a good answer. I came up with $$\Big(\sin(t)\cos^{-1}\big(\tfrac{1}{\sqrt{3}}\big), \sin(t)\sin^{-1}\big(\tfrac{1}{\sqrt{3}}\big), (\text{some linear function})\Big)$$

First $r=1$.

Next we get $(\frac1{\sqrt3},\frac1{\sqrt3},\frac1{\sqrt3})+\cos t\cdot\frac1{\sqrt5} (2,0,1)+\sin t\cdot \frac1{\sqrt{30}}(-1,5,2)$ by looking at this answer.