Introduction
The cross product (also known as the vector product) is an operation on two vectors in three-dimensional space. It results in a vector that is orthogonal (perpendicular) to both of the original vectors, with a magnitude equal to the area of the parallelogram that the vectors span.
Basics of the Cross Product
Given two vectors and in three-dimensional space, their cross product is given by:
The resulting vector is orthogonal to both and , and its magnitude equals the area of the parallelogram spanned by and .
The cross product can also be written as a matrix determinant as follows:
This representation is identical the the earlier equation. However, it is often useful to avoid having to memorize the longer form, so long as one knows how to take a determinant.
Properties of the Cross Product
The cross product operation has several key properties:
- Anticommutativity:
- Distributivity over vector addition:
- Scalar multiplication:
- The cross product of a vector with itself is the zero vector:
Geometric Interpretation
The cross product can be understood geometrically: the magnitude of the cross product equals the area of the parallelogram with sides and . In terms of magnitudes and the angle between and , this can be written as:
The direction of is given by the right-hand rule.
Cross Product in Physics
In physics, the cross product frequently appears in the context of rotational dynamics. For example, the torque exerted by a force applied at a point with position vector relative to the axis of rotation is given by the cross product:
The cross product is also fundamental in defining the magnetic force on a moving charged particle in a magnetic field.
Conclusion
The cross product is a key tool in mathematics and physics for operations in three dimensions. With its unique properties and geometric interpretation, it helps to solve and visualize a wide range of problems, especially those involving rotations and angular momentum.
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