Tru Physics

Cross Product

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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 \vec{a} = (a_1, a_2, a_3) and \vec{b} = (b_1, b_2, b_3) in three-dimensional space, their cross product \vec{a} \times \vec{b} is given by:

\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)

The resulting vector \vec{a} \times \vec{b} is orthogonal to both \vec{a} and \vec{b}, and its magnitude ||\vec{a} \times \vec{b}|| equals the area of the parallelogram spanned by \vec{a} and \vec{b}.

The cross product can also be written as a matrix determinant as follows:

\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

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 3\times 3 determinant.

Properties of the Cross Product

The cross product operation has several key properties:

  1. Anticommutativity: \vec{a} \times \vec{b} = -\vec{b} \times \vec{a}
  2. Distributivity over vector addition: \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}
  3. Scalar multiplication: (k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b}) = \vec{a} \times (k\vec{b})
  4. The cross product of a vector with itself is the zero vector: \vec{a} \times \vec{a} = \vec{0}

Geometric Interpretation

The cross product can be understood geometrically: the magnitude of the cross product ||\vec{a} \times \vec{b}|| equals the area of the parallelogram with sides \vec{a} and \vec{b}. In terms of magnitudes and the angle \theta between \vec{a} and \vec{b}, this can be written as:

||\vec{a} \times \vec{b}|| = ||\vec{a}|| \cdot ||\vec{b}|| \cdot \sin(\theta)

The direction of \vec{a} \times \vec{b} 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 \vec{\tau} exerted by a force \vec{F} applied at a point with position vector \vec{r} relative to the axis of rotation is given by the cross product:

\vec{\tau} = \vec{r} \times \vec{F}

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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