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Let $A, B, C$ be vectors in $\mathbb R^2$. I want to show that the set $\{A+tB+t^2C\mid t\in\mathbb R\}$ defines a parabola in $\mathbb R^2$, but I'm having a hard time doing so, since I can't solve for one coordinate in terms of the other.

Michael Hardy
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Nishant
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    It's not true. If $A=B=C=(1,1)$ then you have the line $y=x$. Or if in general if A,B,C are proportional then you will get a line through the origin. – David P Oct 17 '14 at 21:31
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    @David, in that case you only get a closed half-line starting at $(\frac34,\frac34)$. More genreally, if $B$ and $C$ are linearly dependent, then you get a line, a closed ray, or a point. If $B$ and $C$ are linearly independent you get a parabola. – hmakholm left over Monica Oct 17 '14 at 21:33
  • @HenningMakholm true. – David P Oct 17 '14 at 21:34
  • Yeah, I guess I should have ruled out trivial cases... – Nishant Oct 17 '14 at 21:43

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Rotate the coordinate system such that $C$ is parallel to the $y$-axis.

Then, given $x$ you can solve for $t$ (uniquely) and find the corresponding $y$ as a second-degree polynomial in $x$.

hmakholm left over Monica
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