Right-Hand Rule

The right-hand rule is used to determine whether a coordinate system is right-handed, the positive direction of rotation around an axis in such a system, and the direction of the vector returned by the cross product in such a system.

The reason the right-hand rule exists is because there are two ways to construct a 3D coordinate system. A quick way to visualize both types of coordinate systems is to draw a coordinate system, label the axes, and reflect that coordinate system using a mirror.

Two xyz coordinate systems separated by a dashed mirror line; z points up on both, but the left system has x forward-right and y forward-left, while the mirrored right system has them swapped

These two coordinate systems are distinct from each other. In other words, there is no way to rotate one of these coordinate systems so that their positive axes all line up and point the same direction. They are called left-handed and right-handed coordinate systems, and the right-hand rule is the test that tells them apart.

Determining the Handedness of a Coordinate System

To determine if a coordinate system is right-handed, point the index finger of your right hand in the direction of the positive -axis and your middle finger in the direction of the positive -axis. If your thumb points in the same direction as the positive -axis, the coordinate system is right-handed.

Right hand with index finger pointing along the positive x-axis, middle finger along the positive y-axis, and thumb along the positive z-axis

Once you’ve established the handedness of the system, the rule has two more uses: finding the positive direction of rotation around an axis, and finding the direction of the vector returned by the cross product.

Positive Direction of Rotation

In a right-handed coordinate system, the right-hand rule determines the positive direction of rotation around an axis. Stick the thumb of your right hand in the direction of the axis and curl your fingers. The direction your fingers curl is the positive direction of rotation.

Right hand with thumb extended along an axis and fingers curling in the direction of positive rotation around it

Direction of the Cross Product

In a right-handed coordinate system, the right-hand rule also determines the direction of the vector returned by the cross product. Point the index finger of your right hand in the direction of the first vector and your middle finger in the direction of the second vector. Your thumb then points in the direction of the cross product. This is the same hand position as the handedness test, with the axes relabeled as vectors.

Right hand with index finger pointing along vector v, middle finger along vector w, and thumb along the cross product v cross w

Practice Questions

Three-dimensional coordinate system on a grid plane where the positive x-axis points back to the right, the positive y-axis points forward toward the viewer, and the positive z-axis points up
In the image above, is the coordinate system right-handed or left-handed?
Three-dimensional coordinate system with x, y, and z axes. A straight arrow points diagonally upward between the positive y-axis and positive z-axis, representing an axis of rotation. A curved arrow wraps around it showing a direction of rotation.
In the image above, does the curved arrow represent the positive direction of rotation around the straight axis?
Right-handed three-dimensional coordinate system showing vector v pointing up and back, vector w pointing down, the parallelogram they span, and a labeled arrow for v cross w extending perpendicular to that parallelogram into the upper-right region of the figure.
In the image above, does the arrow labeled correctly show the direction of the cross product of and ?

References

  1. Cross Products | 3Blue1Brown
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