Prove that the unit circle, for an inner product on lR^2 is defined as the set of all vectors of unit length ||v|| = 1, of the non-standard inner product [tex]v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2[/tex] is an ellipse.(adsbygoogle = window.adsbygoogle || []).push({});

I know that norm squared will be [tex](v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2) (v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2)[/tex], but I don't really want to multiply that all out to show that it looks like an ellipse. Is there a better way, maybe manipulating the inner product somehow?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Weighted unit circle

**Physics Forums | Science Articles, Homework Help, Discussion**