Full disclosure, this is for a Calculus III graded homework set--though we are allowed to use any resources available to complete it.
I feel I have a good understanding of space curves, though my understanding of the equation of planes leaves a lot to be desired, as well as transformations on space curves.
The full wording of the question is this: Find the vector equation of the 3-dimensional space curve r(t) that represents an ellipse whose center is at (7,3,-5) that lines in the plane 2x-y+z=6. Also verify that the ellipse lies in the given plane(I have no problem with the last part)
I know that the equation of the ellipse will have to be of the form: $f(a\cos(t),b\sin(t),z)$. Technically, the a and b are unnecessary because a circle is just a special case of an ellipse, but I'd prefer to find it in general and then just plug in some constants after the fact--doing a circle is cheap, in my opinion.
I know that the plane is defined by its normal vector, <2,-1,1>, and that D=6 defines its shift from the origin. My issue I'm not sure to incorporate this shift into the equation of the ellipse, and I'm not quite sure how to incorporate the normal vector. I'm planning on finding the vector equation for an ellipse with the given center and then "forcing" it to always be orthogonal to the normal vector--is this the correct approach? I can do most of this on my own, but I'm not sure of where to start.
Thank you in advance for your help.