Right-hand rule

From Free net encyclopedia

The right hand rule is also an algorithm used to solve mazes, where a person keeps their right hand on a wall at all times while moving forward, because the walls along the path to the exit of the maze form one surface.

In mathematics and physics, the right-hand rule is a convention for determining relative directions of certain vectors.

In fact, there are two closely related right-hand rules.

1 Direction associated with an ordered pair of directions
2 Direction associated with a rotation
3 Applications of the right-hand rule
4 Left-hand rule
5 See also
6 External links

[edit]

Direction associated with an ordered pair of directions

Image:Cartesian coordinate system handedness.svg The first of these occurs in situations in which a non-commutative operation must be performed on two directions a and b (in a three-dimensional space) that constructs a direction c perpendicular to both a and b. There are in fact two such directions. The right-hand rule imposes the following procedure for choosing one of the two directions.

First, the hand is held flat and positioned so that the fingers are aligned with a. Then, the hand is rotated about the forearm so that the fingers curl inward toward b. The thumb indicates c.

There is also an alternative technique. First, the forefinger of the right hand is pointed directly forward, and the entire hand positioned so that the forefinger is aligned with a. Then, the middle finger is turned inward (toward the palm), and the hand is turned about the axis defined by a so that the middle finger aligns with b. The thumb indicates c. Alternatively, consider a person standing at the origin facing along a and the direction of their head (in their own frame) being b. Then their right hand gives c

In electromagnetism (physics), yet a third technique is used for determining the direction of a force due to a charge moving in a magnetic field (see Lorentz force). Using the right hand, the fingers represent the direction of the magnetic field, the thumb is the direction of the current, and the resulting force is in the direction the palm appears to "push".

[edit]

Direction associated with a rotation

The other form of the right-hand rule occurs in situations where a direction c must be determined based on a rotational direction, or vice versa. In this case, the fingers of the right hand are curled in the rotational direction, and the thumb indicates c.

Correspondingly:

moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like a screw
leftward direction is associated with the rotation of the wheels of a vehicle moving forward

The relation with the previous section is established by associating with directions a and b the rotation over the shortest angle from a to b.

[edit]

Applications of the right-hand rule

Perhaps the most fundamental application of the right-hand rule is the Cartesian coordinate system, where the first form is used to position the z-axis once the x- and y-axes have been determined.

The first form of the rule is also used to determine the direction of the cross product of two vectors. This leads to widespread use in physics, wherever the cross product occurs. A list of physical quantities whose directions are related by the right-hand rule is given below. (Some of these are related to cross products only indirectly, and use the second form.)

The angular velocity of a rotating object and the rotational velocity of any point on the object
A torque, the force that causes it, and the position of the point of application of the force
A magnetic field, the position of the point where it is determined, and the electric current (or change in electric flux) that causes it
A magnetic field in a coil of wire and the electric current in the wire
The force of a magnetic field on an object, the magnetic field itself, and the velocity of the object