Gravity Probe B Notes: Projecting Vectors via the Dot Product and the Importance of High School Trig
I'm in the process of reading Schiff's Gravity Probe B inception paper[1]. Gravity Probe B was the satellite borne experiment that detected the Earth's gravitomagnetic field, but that's not what I'll be talking about today. This post is more about a math trick/pattern. It's a mathematical pattern that comes up pretty frequently in physics, so I figured it was worth a few notes here. The first picture below shows the equation for the torque on a spinning object due to a spherical source of gravity, (like the Earth), with a bit of its attendant explanation by Schiff. My notes can be seen to the left:
The cool part I'm going to focus on today is one of the smallest expressions within the rather ginormous equation 3, (also shown in picture 2):
$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r^2$$
I've run into structures like this in the past and it took me awhile to realize what they did. Likewise for some of my classmates. The short version of the story is that this operation gives the projection of the omega vector along the r vector's direction. It's all done with dot products, and consequently, high school trigonometry. Here's a picture:
We've got an arbitrary omega and r vector. The cosine operation shown gives the horizontal component of the omega vector with respect to the r vector. This is just plain old trig. The vector equation above accomplishes the same thing with a different notation. First, we write down the dot product as, (in the G+ version, the following four equations can be found in the pictures as well),
$$\left(\vec{\omega}\cdot\vec{r}\right) = |\vec{\omega}||\vec{r}|cos\theta$$
and we're most of the way there. The issue is that the dot product carries along a factor equal to the length of the r vector. That's not what we want, so we divide it out giving:
$$\left(\vec{\omega}\cdot\vec{r}\right)/r = |{\omega}|cos\theta$$
and we're done... except we're not. In the expression from Schiff's paper we have a vector that points in the r direction. To get this, we can just multiply the above numeric result times the r vector
$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r = |{\omega}|cos\theta\; \vec{r}$$
The cool part I'm going to focus on today is one of the smallest expressions within the rather ginormous equation 3, (also shown in picture 2):
$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r^2$$
I've run into structures like this in the past and it took me awhile to realize what they did. Likewise for some of my classmates. The short version of the story is that this operation gives the projection of the omega vector along the r vector's direction. It's all done with dot products, and consequently, high school trigonometry. Here's a picture:
We've got an arbitrary omega and r vector. The cosine operation shown gives the horizontal component of the omega vector with respect to the r vector. This is just plain old trig. The vector equation above accomplishes the same thing with a different notation. First, we write down the dot product as, (in the G+ version, the following four equations can be found in the pictures as well),
$$\left(\vec{\omega}\cdot\vec{r}\right) = |\vec{\omega}||\vec{r}|cos\theta$$
and we're most of the way there. The issue is that the dot product carries along a factor equal to the length of the r vector. That's not what we want, so we divide it out giving:
$$\left(\vec{\omega}\cdot\vec{r}\right)/r = |{\omega}|cos\theta$$
and we're done... except we're not. In the expression from Schiff's paper we have a vector that points in the r direction. To get this, we can just multiply the above numeric result times the r vector
$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r = |{\omega}|cos\theta\; \vec{r}$$
There's only one last issue left. The r vector points in the correct direction, but it has the magnitude of r attached to it. Consequently, when we multiplied by it, we wound up with our result being r times bigger than it should be again. So... we just divide out another factor of the length of the r vector to get
$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r ^2= |{\omega}|cos\theta\; \hat{r}$$
and we finally have the component of the omega vector pointing in the r direction! The r vector with a hat instead of an arrow over it indicates that the vector points in the correct direction, but that its length is one.
References:
1. Schiff's paper describing the original ideas behind Gravity Probe B, the satellite experiment that detected the Earths' gravitomagnetic field
Schiff, L.I., "Possible New Experimental Test of General Relativity Theory", Physical Review Letters, 4, (1960), 215
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