Hello, 1005!
That's very perceptive . . . good for you!
You may not need convincing, but I agree with you . . .
We have a circle of radius 2 centered at (1,2,5) in a plane parallel to the yzplane.
Are both of these answers correct?
. . $\displaystyle \begin{array}{ccc}r(t) &=& (1, \:2 + 2\cos t, \:5 + 2\sin t) \\ r(t) &=& (1, \:2 + 2\sin t, \:5 + 2\cos t)\end{array}$ . . . . Yes!
I cannot think of a reason why both would not be correct.
The only difference I see is what point $\displaystyle t = a$ represents.
I ask because the back of my book only gives the first. For reference, my coordinate system is set up like this: Code:
z




+      y
/
/
/
/
x
The positive $\displaystyle z$axis is upward, the positive $\displaystyle y$axis is to the right,
. . and the positive $\displaystyle x$axis "comes out of the screen."
The plane of the circle is parallel to the "back wall,"
. . so the circle is facing us.
With your first equation, $\displaystyle t \:=\:0$ starts the circle at 3:00
. . and $\displaystyle t \to 2\pi$ generates the circle counterclockwise.
With your second equation, $\displaystyle t \:=\:0$ starts the circle at 12:00
. . and $\displaystyle t \to 2\pi$ generates the circle clockwise.
Either way, we get the same circle.