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The right hand rule (RHR) is important for understanding the directions of the vectors involved in a cross product. Let me teach you some techniques and tricks that work. 😊
RHR for the Coordinate Directions
Coordinate Systems are Right-Handed.
The three-dimensional coordinate systems that we use in mathematics actually follow the RHR. For example, the rectangular (Cartesian) coordinate system \(xyz\) follows this relationship for their unit vectors:
\[\begin{equation} \label{eqn:rhrxyz} \hat{i} \times \hat{j} = \hat{k}\end{equation}\]
Therefore, in understanding the RHR for cross products, we can refer to the right-handedness of coordinate systems. The Cartesian coordinate system is illustrated in Figure 1 below using the common directions we associate with them: \(+x\) is rightward, \(+y\) is upward, and \(+z\) is outward (by the way, the circle with a dot inside means outward, while the circle with an "X" inside means inward). For our purpose, we use the color codes: red for +x, green for +y, and blue for +z.
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| Figure 1. The Cartesian coordinate system. |
The Cartesian Coordinate System is embedded on your left hand (yes, you read that right [pun intended 😂]).
Interestingly, did you know that the simplest way to recreate the coordinate system shown in Figure 1 is through the fingers of your LEFT hand?
Let +x be your middle finger.
Let +y be your index finger.
Let +z be your thumb.
In this example, if you place your middle finger rightward and your index finger upward, recreating the directions for +x and +y, respectively, as indicated above, your thumb will naturally point outward in the indicated direction for +z.
[When I teach this in an actual class, I used to joke that all who understood the lesson should raise their left hand with the fingers arranged this way. This is our three-dimensional "OK" sign.]
Notice that if you use your right hand, you will not be able to recreate Figure 1. At least not simply.
The RHR will naturally follow.
This time, we can verify that the cross product given by Equation (\(\ref{eqn:rhrxyz}\)) follows the RHR. Just follow these steps using your RIGHT hand this time (ideally, with your left hand still recreating the Cartesian coordinate system above):
STEP 1: Point your four fingers to the direction of the first vector.
Here, the first vector in the cross product is \(\hat{i}\), which is rightward.
The thing is, if I point my four fingers to the right, there are a number of possible ways I can do it. So this step should be accompanied by a second one:
STEP 2: Your palm should open towards the direction of the second vector.
Here, the second vector in the cross product is \(\hat{j}\), which is upward.
This time, the direction is more definite. While your four fingers are rightward (STEP 1), your palm opens upward (STEP 2).
STEP 3: Curl your four fingers towards the direction of the second vector. The direction of the third vector is where the thumb points at.
Here, the second vector in the cross product is \(\hat{k}\), which is outward.
[I noticed that this "curling" is the part that is being emphasized by our book, but doing STEPS 1 and 2 will make the job more accurate and easier.]
Generalizing RHR for Cross Products
The cross product appears quite often in physics.
The general equation for the cross product of two vectors \(\vec{A}\) and \(\vec{B}\) is:
\[\begin{equation} \label{eqn:cross} \vec{A} \times \vec{B} = \vec{C}\end{equation}\]
where the vector \(\vec{C}\) is the result of the cross product.
We already encountered a cross product in Mechanics, when we solved for the torque \(\vec{\tau}\) resulting from the action of a force \(\vec{F}\) at a moment arm \(\vec{r}\) away from a pivot point:
\[\begin{equation}\label{eqn:torque} \vec{r} \times \vec{F} = \vec{\tau} \end{equation}\]
Here, we will use the cross product for getting the magnetic force \(\vec{F}_m\) acting on a moving charge \(q\) with velocity \(\vec{v}\) in a region with magnetic field \(\vec{B}\):
\[\begin{equation}\label{eqn:fm} q\vec{v} \times \vec{B} = \vec{F}_m\end{equation}\]
We can generalize the directions.
As we did previously, we can let the following color codes apply:
The first vector \(\vec{A}\) is red.
The second vector \(\vec{B}\) is green.
The third vector \(\vec{C}\) is blue.
(or, in our particular example for the magnetic force, \(q \vec{v}\), \(\vec{B}\), and \(\vec{F}_m\), respectively.)
And the left hand representation.
And, by extension, the directions of the left hand as indicated earlier:
Let \(\vec{A}\) be your middle finger.
Let \(\vec{B}\) be your index finger.
Let \(\vec{C}\) be your thumb.
Now, here's the thing: these vectors, unlike the unit vectors with fixed directions, can be along any coordinate directions! This time, we can use the left hand to find the resulting directions for the vectors, as long as any two directions are known.
Example 1: Consider the case when \(\vec{A}\) is along +x, and \(\vec{B}\) is along +y. Where is \(\vec{C}\)? (See Figure 2; pardon my chubby fingers.)
Let \(\vec{A}\) be your middle finger. : Point it rightwards (+x)
Let \(\vec{B}\) be your index finger. : Point it upwards (+y)
Let \(\vec{C}\) be your thumb. : Verify that it will point... (OUTWARD, +z)
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| Figure 2. Illustrating Example 1. Colors denote the vectors \(\vec{A}\), \(\vec{B}\), \(\vec{C}\). |
Example 2: Consider the case when \(\vec{A}\) is along +y, and \(\vec{B}\) is along -z. Where is \(\vec{C}\)? (See Figure 3; sorry if it doesn't look particularly nice, I am not giving anybody the finger huhu...)
Let \(\vec{A}\) be your middle finger. : Point it upwards (+y)
Let \(\vec{B}\) be your index finger. : Point it inwards (or away from you, -z)
Let \(\vec{C}\) be your thumb. : Verify that it will point... (LEFTWARD, -x)
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| Figure 3. Illustrating Example 2. Colors denote the vectors \(\vec{A}\), \(\vec{B}\), \(\vec{C}\). |
Example 3: Consider the case when \(\vec{A}\) is along +y, and \(\vec{C}\) is along -z. Where is \(\vec{B}\)? (See Figure 4; sorry again for the placement of the finger huhu...)
Let \(\vec{A}\) be your middle finger. : Point it upwards (+y)
Let \(\vec{C}\) be your thumb. : Point it inwards (or away from you, -z)
Let \(\vec{B}\) be your index finger. : Verify that it will point... (RIGHTWARD, +x)
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| Figure 4. Illustrating Example 3. Colors denote the vectors \(\vec{A}\), \(\vec{B}\), \(\vec{C}\). |
See the advantage of this approach? Notice that this gives you the correct answer even if the first, second, or third vector is unknown. You might have to twist your hand a bit for some cases, but it sure is worth it, because, done correctly, it will give you the correct answer for any given set of vectors.
Now let us do some exercise, shall we? This may involve an actual exercise for your fingers. This will not be graded, and meant for pure fun! You may consult with your classmates to compare your answers.
Exercise 1: If \(\vec{A}\) is along +z, and \(\vec{B}\) is along -x, where is \(\vec{C}\)?
Exercise 2: If \(\vec{B}\) is along -y, and \(\vec{C}\) is along -z, where is \(\vec{A}\)?
Exercise 3: If \(\vec{A}\) is along +x, and \(\vec{C}\) is along -y, where is \(\vec{B}\)?
But, hey, isn't this tutorial about the right hand rule?
Indeed it is, and you can, in fact, answer the given examples and exercises by generalizing the RHR given earlier:
STEP 1: Point your four fingers to the direction of the first vector.
STEP 2: Your palm should open towards the direction of the second vector.
STEP 3: Curl your four fingers towards the direction of the second vector. The direction of the third vector is where the thumb points at.
Now these steps have an inherent advantage over the left hand representation earlier. There are cases wherein the vectors are not oriented along the \(x\), \(y\), or \(z\) axes. Consider the following example:
Example 4. The vector \(\vec{A}\) is oriented \(45^\circ\) cc from the +x-axis, while \(\vec{B}\) is oriented \(30^\circ\) cc from the -y-axis. Where will \(\vec{C}\) be oriented?
This time, you can follow the steps outlined above (don't forget STEP 2!) and verify that the answer is INWARD, -z.
To verify, use these steps for all the given examples and exercises and verify that you will get the same answer. █
For next meeting, we will specifically look at Equation (\(\ref{eqn:fm}\)).
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