Right Hand Rule

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 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.

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: 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.
• 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.

To find the magnetic field induced by a wire, you can do the same thing, point your thumb in the direction of current and your fingers show the magnetic field.

Counterclockwise example: 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. 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 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.

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

he right-hand rule developed in the 1800s as scientists tried to standardize how to describe directions in electricity, magnetism, and rotational motion. After Orsted and Ampère discovered that electric currents create magnetic fields, a consistent way to define the field’s direction was needed. Maxwell later unified electromagnetism and used a right-handed coordinate system in his equations, solidifying the convention. In the late 1800s, Gibbs and Heaviside formally defined the modern cross product, specifying its direction with the right hand, which made the rule a permanent part of vector calculus. Today, the right-hand rule is universally used to assign consistent directions to quantities like 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

References

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

• Cross product visualization
• All Right Hand Rule visualization pictures

All other pictures and videos were taken from Professor Flavio Fenton's Physics 2211 lecture notes. https://en.wikipedia.org/wiki/Right-hand_rule