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This topic covers Right Hand Rule.  
This topic covers Right Hand Rule.  
Claimed by: Jake Vollkommer
Claimed by: Bhavin Shah Spring 2025
<!-- Whoever edits this: please make sure the reader knows that right hand rule refers to cross products in general, not just angular momentum-->


==The Main Idea==
==The Main Idea==
The magnitude of the translational angular momentum doesn't tell us everything about the motion of an object. For example, to describe the motion of Earth's orbit if you are standing on the positive z axis looking toward the origin, the Earth is moving counterclockwise in the xy plane. However, this statement cannot be used mathematically. You can't add "counterclockwise in the xy plane" to "clockwise in the yz plane." Instead, we can use the right hand rule to describe the direction of angular momentum as a vector. The direction can be specified like this:
The right-hand rule is not just a mnemonic that teachers tell you to use—it’s a geometric consequence of how three-dimensional space is defined. Many physical quantities—such as angular momentum, torque, magnetic forces, and rotational motion—are fundamentally tied to the cross product, and the right-hand rule is what gives those quantities a consistent and meaningful direction. Understanding the intuition behind the RHR, combined with practice, allows us as students to move through problems more quickly on tests. By understanding the direction of the resultant vector, we can often use it to guide our work and even use it as a proxy to check whether our answers make sense.
* The plane in which both the position vector from the two objects and the momentum vector for the object of interest lie can be indicated by a unit vector perpendicular to that plane.  
* The direction of motion within the plane (clockwise or counterclockwise) can be indicated by establishing a right hand rule (RHR) for this unit vector.


===A Mathematical Model===
==Why We Need the Right-Hand Rule==
The direction of the angular momentum can also be solved through calculating the cross product of the '''r''' and '''p''' vectors.  
When two vectors lie in a plane—say, the position vector r of a particle and its momentum vector p—their combination often represents something that is not confined to that plane. For example:
The cross product of two vectors can be solved as so:  


[[File:crossproduct2.jpg]]
The angular momentum of a particle does not lie in the plane of motion. The magnetic force on a charge depends on a cross product and points perpendicular to the plane of the velocity and magnetic field. The torque exerted on an object is also perpendicular to the plane of applied force and lever arm. The cross product gives a vector that sticks straight out of the plane formed by the two original vectors. But there are two possible directions it could point—one on each side of the plane. The right-hand rule is the convention we use to pick the correct, consistent direction every time. Without this rule, the direction of quantities like torque, angular momentum, or magnetic force would be ambiguous.


[[File:Cross product.jpg]]
===A Mathematical and Computational Model===
The direction of any quantity defined by a cross product—such as torque, magnetic force, or angular momentum—can be found by computing the cross product of two vectors. For any vectors
𝑎
a and
𝑏
b, the cross product
 
[[File:crossproduct-vis.png]]
 
To calculate the cross product from a matrix, multiply each component by its corresponding determinant. Use this visual representation to help you.
[[File:crossproduct-visual.png]]


The first row is the standard basis vectors and must appear in the order given here.  The second row is the components of '''r''' and the third row is the components of '''p'''. First, the terms alternate in sign and notice that the 2x2 is missing the column below the standard basis vector that multiplies it as well as the row of standard basis vectors. To solve for the "i" component, you use the expression: bf - ce. Do the same for the "j" and "k" components, and this will give you your vector.  Your vector should be <bf-ce, af-cd, ae-bd>.


For more on the explanation of how to calculate cross product, visit this website: https://www.mathsisfun.com/algebra/vectors-cross-product.html
For more on the explanation of how to calculate cross product, visit this website: https://www.mathsisfun.com/algebra/vectors-cross-product.html
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[[File:problem1.jpg]]
[[File:problem1.jpg]]
For a computational model i have attached the trinket code below which provides a model of the the right hand rule cross-product.
https://trinket.io/glowscript/31d0f9ad9e


Answer:
Answer:
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Now, take your right hand and point your index finger forward (the direction your arm faces), point your thumb up, and point your middle finger perpendicular to your index finger.  
Now, take your right hand and point your index finger forward (the direction your arm faces), point your thumb up, and point your middle finger perpendicular to your index finger.  
Your thumb is the z-axis and points in the positive z-direction. Your index and middle fingers represent the x-y plane.  
Your thumb is the z-axis and points in the positive z-direction. Your index and middle fingers represent the x-y plane.  
Your hand should look something like this:    Now visualize the x-y-z axes over your hand like this:
 
Your hand should look something like this:


[[File:right-hand250.png]]
[[File:right-hand250.png]]
Then you can visualize the x-y-z axes over your hand like this:
[[File:right-hand-directions250.png]]
[[File:right-hand-directions250.png]]


Realize that you can rotate the x-y plane along the z-axis.  
Realize that you can rotate the x-y plane along the z-axis.  
For example, if you were to rotate the original x-y plane 90 degrees clockwise, you would see this:
For example, if you were to rotate the original x-y plane in 90 degree increments clockwise, you would see this:
 
[[File:xyz1rot500.png]]
 
Rotate the plane once more 90 degrees clockwise:


[[File:xyz2rot500.png]]
[[File:xyzog250f.png]]
[[File:xyz1rot250h.png]]
[[File:xyz2rot250h.png]]


Now, as you can see, your right hand indeed represents the entire x-y-z space.
Now, as you can see, your right hand indeed represents the entire x-y-z space.
Line 52: Line 66:
==Performing the Right Hand Rule==
==Performing the Right Hand Rule==


* Right Hand Rule: Using your right arm, point your arm to represent the '''r''' vector. Now turn your palm in the direction of the momentum or '''p''' vector. Curl your fingers in that direction of the momentum, and extend your thumb outward. The unit vector representing the direction of the angular momentum is defined to point in the direction of your thumb.
Right-Hand Rule:
*Hint: If the rotational motion is counterclockwise, your right thumb, therefore the unit vector, will point out of the plane. If the rotational motion is clockwise, the unit vector will point into the plane.  
Extend your right arm so that it points in the direction of the r vector. Rotate your wrist so that your palm can curl toward the direction of the p (momentum) vector. Curl your fingers in the direction of p, and extend your thumb. The thumb now points in the direction of the unit vector that represents the angular momentum.
 
Hint:
If the rotation is counterclockwise, your right thumb—and therefore the angular momentum vector—points out of the plane (+z direction).
If the rotation is clockwise, the vector points into the plane (–z direction).
 
Right-Hand Rule (for magnetic fields around a current-carrying wire):
Point your thumb in the direction of the current. Your curled fingers naturally wrap around the wire in the direction of the magnetic field lines.


Counterclockwise example:
Counterclockwise example:
 
When the direction from r to p is counterclockwise, your curled fingers match that motion, and your thumb points upward in the +z direction. This confirms that the angular momentum vector points out of the page.
[[File:counterclockwise-example500.png]]
[[File:counterclockwise-example500.png]]
Notice your thumb points up in the +z direction when the direction from '''r''' to '''p''' is counterclockwise.


If the '''p''' vector is in the other direction relative to '''r''', you would need to turn your hand upside down to curl your fingers towards it.
If the '''p''' vector is in the other direction relative to '''r''', you would need to turn your hand upside down to curl your fingers towards it.
In this case, your thumb points down in the -z direction!


[[File:clockwise1.png]]
[[File:clockwise-ex1.png]]


In this case, your thumb points down in the -z direction!


This situation would also make your thumb point in the -z direction (Notice it is just the first image rotated 180 degrees).
This situation would also make your thumb point in the -z direction (Notice it is just the first image rotated 180 degrees).


[[File:clockwise2.png]]
[[File:clockwise-ex2.png]]


==Examples==
==Examples==
Line 93: Line 112:


==Connectedness==
==Connectedness==
The Right Hand Rule is a very essential component of solving cross products, such as in angular momentum problems. The concept is usually very tricky to understand, so I wanted to give my best insight on what helped me understand exactly how and why it works.  
The Right Hand Rule is a very essential component of solving cross products, such as in Hall Effect, Angular Momentum, Lorentz force and more. The concept is usually very tricky to understand, so I wanted to give my best insight on what helped me understand exactly how and why it works. It also provides a proxy for right answers whcih will be useful in a testing environment.
 
 
Here are some interesting videos about the conservation of angular momentum:
 
 
https://www.youtube.com/watch?v=Aw5i994n2bw&feature=youtu.be
 
https://www.youtube.com/watch?v=OKbawIq3w7U


==History==
==History==
 
The right-hand rule emerged in the 1800s as scientists sought a consistent method for describing directions in electricity, magnetism, and rotational motion. After Ørsted and Ampère showed that electric currents generate magnetic fields, physicists needed a standardized way to define the direction of those fields. James Clerk Maxwell later unified electricity and magnetism and adopted a right-handed coordinate system in his equations, helping solidify the convention. By the late 19th century, Josiah Willard Gibbs and Oliver Heaviside formally defined the modern vector cross product and specified its direction using the right hand, embedding the rule into vector calculus. Today, the right-hand rule is universally used to assign consistent directions to physical quantities such as torque, angular momentum, and magnetic forces.
I wasn't able to find who was the first to apply the Right Hand Rule to angular momentum and torque. However, it seems that the right hand rule is applied to other aspects of physics
as well. For example, André-Marie Ampère, a French physicist and mathematician, created a right hand rule for circuits and electric currents. This is used when a vector must be defined
to represent the rotation of a body, a magnetic field, or a fluid. It reveals a connection between the current and the magnetic field lines in the magnetic field that the current created.  
This right hand rule works exactly the same way as the one I have described above.


== See also ==
== See also ==
Line 118: Line 125:


[http://mathworld.wolfram.com/Right-HandRule.html Wolfram MathWorld]
[http://mathworld.wolfram.com/Right-HandRule.html Wolfram MathWorld]
https://trinket.io/glowscript/31d0f9ad9e
https://www.mathsisfun.com/algebra/vectors-cross-product.html
https://teachingandlearninglabs.com/AllHTML/RightHandRule.shtml


==References==
==References==


Right hand rule pictures were taken by me and are of my hand.
The following pictures were created in Adobe Photoshop using pictures taken by me:
All other pictures and videos were taken from Professor Flavio Fenton's Physics 2211 lecture notes.  
*Cross product visualization
*All Right Hand Rule visualization pictures
 
All pictures were taken from lecture slides to refrain from infrangement.
 
https://www.physicsclassroom.com/class/Magnetic-Fields-and-Electromagnetism/Lesson-2-Electromagnetism/Magnetic-Field-of-a-Current-Carrying-Wire
 
https://www.khanacademy.org/test-prep/mcat/physical-processes/magnetism-mcat/a/using-the-right-hand-rule
 
https://www.vecteezy.com/vector-art/25747595-fleming-s-right-hand-rule
 
https://en.wikipedia.org/wiki/Right-hand_rule
https://en.wikipedia.org/wiki/Right-hand_rule

Latest revision as of 23:14, 30 November 2025

This topic covers Right Hand Rule. Claimed by: Bhavin Shah Spring 2025

The Main Idea

The right-hand rule is not just a mnemonic that teachers tell you to use—it’s a geometric consequence of how three-dimensional space is defined. Many physical quantities—such as angular momentum, torque, magnetic forces, and rotational motion—are fundamentally tied to the cross product, and the right-hand rule is what gives those quantities a consistent and meaningful direction. Understanding the intuition behind the RHR, combined with practice, allows us as students to move through problems more quickly on tests. By understanding the direction of the resultant vector, we can often use it to guide our work and even use it as a proxy to check whether our answers make sense.

Why We Need the Right-Hand Rule

When two vectors lie in a plane—say, the position vector r of a particle and its momentum vector p—their combination often represents something that is not confined to that plane. For example:

The angular momentum of a particle does not lie in the plane of motion. The magnetic force on a charge depends on a cross product and points perpendicular to the plane of the velocity and magnetic field. The torque exerted on an object is also perpendicular to the plane of applied force and lever arm. The cross product gives a vector that sticks straight out of the plane formed by the two original vectors. But there are two possible directions it could point—one on each side of the plane. The right-hand rule is the convention we use to pick the correct, consistent direction every time. Without this rule, the direction of quantities like torque, angular momentum, or magnetic force would be ambiguous.

A Mathematical and Computational Model

The direction of any quantity defined by a cross product—such as torque, magnetic force, or angular momentum—can be found by computing the cross product of two vectors. For any vectors 𝑎 a and 𝑏 b, the cross product

To calculate the cross product from a matrix, multiply each component by its corresponding determinant. Use this visual representation to help you.


For more on the explanation of how to calculate cross product, visit this website: https://www.mathsisfun.com/algebra/vectors-cross-product.html

Here is an example problem. Solve first using the right hand rule, and then solve mathematically with the cross product.

For a computational model i have attached the trinket code below which provides a model of the the right hand rule cross-product.

https://trinket.io/glowscript/31d0f9ad9e


Answer: The cross product should be <-12, 0, 0> So the vector is in the -x direction, which the right hand rule also tells us.

Visualization

You can use your right hand to visualize the intuition of the Right Hand Rule. Consider the x-y-z space drawn with these axes from the origin:

Now, take your right hand and point your index finger forward (the direction your arm faces), point your thumb up, and point your middle finger perpendicular to your index finger. Your thumb is the z-axis and points in the positive z-direction. Your index and middle fingers represent the x-y plane.

Your hand should look something like this:

Then you can visualize the x-y-z axes over your hand like this:

Realize that you can rotate the x-y plane along the z-axis. For example, if you were to rotate the original x-y plane in 90 degree increments clockwise, you would see this:

Now, as you can see, your right hand indeed represents the entire x-y-z space.

Performing the Right Hand Rule

Right-Hand Rule: Extend your right arm so that it points in the direction of the r vector. Rotate your wrist so that your palm can curl toward the direction of the p (momentum) vector. Curl your fingers in the direction of p, and extend your thumb. The thumb now points in the direction of the unit vector that represents the angular momentum.

Hint: If the rotation is counterclockwise, your right thumb—and therefore the angular momentum vector—points out of the plane (+z direction). If the rotation is clockwise, the vector points into the plane (–z direction).

Right-Hand Rule (for magnetic fields around a current-carrying wire): Point your thumb in the direction of the current. Your curled fingers naturally wrap around the wire in the direction of the magnetic field lines.

Counterclockwise example: When the direction from r to p is counterclockwise, your curled fingers match that motion, and your thumb points upward in the +z direction. This confirms that the angular momentum vector points out of the page.

If the p vector is in the other direction relative to r, you would need to turn your hand upside down to curl your fingers towards it. In this case, your thumb points down in the -z direction!


This situation would also make your thumb point in the -z direction (Notice it is just the first image rotated 180 degrees).

Examples

Ans: 4

Ans: 3

Ans: 3

Ans: L(A)=L(B)=L(H) = <0, 0, -30> L(G)=L(C) = <0, 0, 0> L(D)=L(E)=L(F) = <0, 0, +50>

Connectedness

The Right Hand Rule is a very essential component of solving cross products, such as in Hall Effect, Angular Momentum, Lorentz force and more. The concept is usually very tricky to understand, so I wanted to give my best insight on what helped me understand exactly how and why it works. It also provides a proxy for right answers whcih will be useful in a testing environment.

History

The right-hand rule emerged in the 1800s as scientists sought a consistent method for describing directions in electricity, magnetism, and rotational motion. After Ørsted and Ampère showed that electric currents generate magnetic fields, physicists needed a standardized way to define the direction of those fields. James Clerk Maxwell later unified electricity and magnetism and adopted a right-handed coordinate system in his equations, helping solidify the convention. By the late 19th century, Josiah Willard Gibbs and Oliver Heaviside formally defined the modern vector cross product and specified its direction using the right hand, embedding the rule into vector calculus. Today, the right-hand rule is universally used to assign consistent directions to physical quantities such as torque, angular momentum, and magnetic forces.

See also

The Right Hand Rule is a crucial part in solving Angular Momentum and Torque problems.

External links

Khan Academy

Wolfram MathWorld

https://trinket.io/glowscript/31d0f9ad9e

https://www.mathsisfun.com/algebra/vectors-cross-product.html

https://teachingandlearninglabs.com/AllHTML/RightHandRule.shtml


References

The following pictures were created in Adobe Photoshop using pictures taken by me:

  • Cross product visualization
  • All Right Hand Rule visualization pictures

All pictures were taken from lecture slides to refrain from infrangement.

https://www.physicsclassroom.com/class/Magnetic-Fields-and-Electromagnetism/Lesson-2-Electromagnetism/Magnetic-Field-of-a-Current-Carrying-Wire

https://www.khanacademy.org/test-prep/mcat/physical-processes/magnetism-mcat/a/using-the-right-hand-rule

https://www.vecteezy.com/vector-art/25747595-fleming-s-right-hand-rule

https://en.wikipedia.org/wiki/Right-hand_rule

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