Category: 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 u = u_x \hat{x} + u_y \hat{y}+ u_z \hat{z} and v = v_x\hat{x} + v_y \hat{y} + v_z \hat{z}. The coefficients u_i, v_i have units of length. You know the dot product

u \cdot v = u_xv_x + u_yv_y + u_x v_z,

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

u \times v = (u_yv_z - u_zv_y)\hat{x} + (u_zv_x - u_xv_z)\hat{y} + (u_xv_y - u_yv_x)\hat{z}

which is a vector. Its length is the magnitude of the parallelogram defined by u and v. 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?

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Predicting gravitational waves

There has been a lot of excitement recently with the news that physicists at LIGO have directly detected gravitational waves. Many wonderful people have done popular science blog posts or videos about what gravitational waves are, but I haven’t really seen anyone talk much about the mathematics of it. For example, why do Einstein’s equations mean that gravitational waves exist? How did Einstein predict gravitational waves?

It turns out that starting with Einstein’s equations and a few simplifying assumptions, it’s relatively easy to derive the necessity of gravitational waves. That’s what this post will try and do.

Wave2

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Higher Dimensional Ellipsoids

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

\frac{(x - x_0)^2}{r_1} + \frac{(y - y_0)^2}{r_2} = k^2

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?”

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