# Tagged: Geometry

# Unicorns of Geometry: Getting Cross About Products

**Or: Why you should forget about the vector cross product!**

So titled because while both spoken about in tales and legends, neither unicorns nor cross products actually exist. Forget what you think you know about vector geometry!

A question for the beginner physics students: consider two vectors and . The coefficients have units of length. You know the *dot* *product*

,

which is a number, has units of area, and is the square of the length of one vector projected onto the other. You probably also know the other interesting operation, the *cross* *product*

which is a vector. Its length is the magnitude of the parallelogram defined by and . But its coefficients have units of area. How can it still be a vector in the same space as the other two if its coefficients have different units?

# Higher Dimensional Ellipsoids

An ellipse is a set of points in the plane which satisfy a certain equation, namely,

In this sense, an ellipse appears to just be a squashed circle. But there are dozens of ways to think about ellipses – as conic sections, for example – and in this post the idea I’d like to use is that of foci:

You can make an ellipse with two pins and a piece of string tied in a circle. Stick the two pins into a page and loop the string over them. Stretch the string out tight with a pen and draw all around. If you make sure to keep the string taut, you will have drawn an ellipse.

It’s a common enough geometric construction, and I’m sure many readers would know of it. I myself remember my dad showing it to me when I was a little boy. What it tells us is that an ellipse can be though of as the set of points whose total distance from *two* *distinct* points, the foci, is constant (plus the distance between those two points, but that’s always constant).

Naturally, a mathematician asks: “How can I generalise this idea?”