Vector Product of Two Vectors (Cross Product)

Introduction

The vector product, also known as the cross product, is an operation that takes two vectors and returns another vector. This resultant vector is perpendicular to the plane containing the two input vectors. The vector product is widely used in physics and engineering to calculate torque, angular momentum, and areas of geometric shapes.

Definition of the Vector Product

The vector product of two vectors \( \vec{v} \) and \( \vec{w} \) is defined as:

\[ \vec{v} \times \vec{w} = |\vec{v}||\vec{w}|\sin \theta \, \vec{n} \]

Here:

• \( |\vec{v}| \) and \( |\vec{w}| \) are the magnitudes of \( \vec{v} \) and \( \vec{w} \).
• \( \theta \) is the angle between \( \vec{v} \) and \( \vec{w} \) (measured in the plane containing both vectors).
• \( \vec{n} \) is the unit vector perpendicular to the plane containing \( \vec{v} \) and \( \vec{w} \), with its direction determined by the right-hand screw rule.

Key Properties of the Vector Product

The vector product has several important properties:

• \( \vec{v} \times \vec{w} = -\vec{w} \times \vec{v} \) (anti-commutative property).
• \( \vec{u} \times (\vec{v} + \vec{w}) = \vec{u} \times \vec{v} + \vec{u} \times \vec{w} \) (distributive property).
• \( (k\vec{v}) \times \vec{w} = k (\vec{v} \times \vec{w}) \), where \( k \) is a scalar.
• \( \vec{v} \times \vec{v} = 0 \) (the vector product of any vector with itself is zero).

Geometric Interpretation

The magnitude of the vector product, \( |\vec{v} \times \vec{w}| \), represents the area of the parallelogram formed by \( \vec{v} \) and \( \vec{w} \). If the two vectors are:

• Parallel: \( \vec{v} \times \vec{w} = 0 \), since \( \sin \theta = 0 \).
• Perpendicular: \( |\vec{v} \times \vec{w}| = |\vec{v}||\vec{w}| \), since \( \sin \theta = 1 \).

Area of a Triangle Using the Vector Product

The area of a parallelogram formed by two vectors \( \vec{v} \) and \( \vec{w} \) is:

\[ \text{Area of parallelogram} = |\vec{v} \times \vec{w}| \]

The area of the triangle is half the area of the parallelogram:

\[ \text{Area of triangle} = \frac{1}{2} |\vec{v} \times \vec{w}| \]

Right-Hand Screw Rule

The direction of the resultant vector \( \vec{u} \times \vec{v} \) is determined by the right-hand screw rule. To use this rule:

• Point the fingers of your right hand in the direction of \( \vec{u} \).
• Rotate your fingers toward \( \vec{v} \).
• Your thumb points in the direction of the resultant vector \( \vec{u} \times \vec{v} \).

Example Calculations

Example 1: Basic Vector Product

Given \( \vec{v} = [2, 3, 4] \) and \( \vec{w} = [1, 0, -1] \):

The vector product is calculated as:

\[ \vec{v} \times \vec{w} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 1 & 0 & -1 \end{vmatrix} \]

Expanding the determinant:

\[ \vec{v} \times \vec{w} = \hat{i} (3 \cdot -1 – 4 \cdot 0) – \hat{j} (2 \cdot -1 – 4 \cdot 1) + \hat{k} (2 \cdot 0 – 3 \cdot 1) \]

\[ \vec{v} \times \vec{w} = -3\hat{i} + 6\hat{j} – 3\hat{k} \]

Thus, \( \vec{v} \times \vec{w} = [-3, 6, -3] \).

Example 2: Area of a Parallelogram

Given \( \vec{v} = [3, 0, 0] \) and \( \vec{w} = [0, 4, 0] \):

• The magnitude of the vector product is: \[ |\vec{v} \times \vec{w}| = |[3, 0, 0] \times [0, 4, 0]| = |[0, 0, 12]| = 12 \]
• The area of the parallelogram is \( 12 \), and the area of the triangle is \( 6 \).

Applications of the Vector Product

The vector product has practical applications in physics, geometry, and engineering:

• Torque: \( \vec{\tau} = \vec{r} \times \vec{F} \), where \( \vec{r} \) is the position vector and \( \vec{F} \) is the force.
• Angular Momentum: \( \vec{L} = \vec{r} \times \vec{p} \), where \( \vec{p} \) is the linear momentum.
• Area Calculation: Finding areas of parallelograms and triangles.