The vector product, also known as the cross product, is a binary operation on two vectors in three-dimensional space that results in a third vector.
As stated in the reference, if you have two vectors a and b, their vector product is a vector c, denoted as:
c = a × b
Unlike the scalar product (dot product) which yields a scalar value, the vector product yields another vector. This resulting vector c has both a magnitude and a direction.
Key Aspects of the Vector Product
- Magnitude: The magnitude of the resulting vector c is given by the formula derived from the reference:
|c| = |a| |b| sinθ
where |a| is the magnitude of vector a, |b| is the magnitude of vector b, and θ is the angle between vectors a and b (0° ≤ θ ≤ 180°). - Direction: The direction of the resulting vector c is perpendicular to both vector a and vector b. This means c is orthogonal to the plane containing a and b. The specific orientation (which of the two possible perpendicular directions) is determined by the right-hand rule. If you curl the fingers of your right hand from vector a to vector b, your thumb points in the direction of a × b.
Properties
The vector product has several important properties:
- Anti-commutative: a × b = - (b × a). The order matters, and reversing the order reverses the direction of the resulting vector.
- Distributive over addition: a × (b + c) = (a × b) + (a × c)
- Scalar multiple: (ka) × b = k(a × b) = a × (kb), where k is a scalar.
- Cross product with itself: a × a = 0 (the zero vector), since the angle θ between a vector and itself is 0°, and sin(0°) = 0.
- Parallel vectors: If a and b are parallel or anti-parallel, their cross product is the zero vector (0), as the angle θ is 0° or 180°, and sin(0°) = sin(180°) = 0.
Why is it Useful?
The vector product is particularly useful in physics and engineering to represent quantities that are defined by a direction perpendicular to a plane formed by two other vectors. Examples include:
- Torque: The rotational force around an axis (τ = r × F).
- Angular Momentum: A measure of an object's rotational inertia (L = r × p).
- Magnetic Force: The force on a moving charge in a magnetic field (F = qv × B).
Example of Vector Product
Let's consider the standard unit vectors in a 3D Cartesian coordinate system: i (along the x-axis), j (along the y-axis), and k (along the z-axis). These vectors are mutually orthogonal (perpendicular to each other).
-
Vector i × Vector j:
- Magnitude: |i| = 1, |j| = 1. The angle between i and j is 90°.
- |i × j| = |i| |j| sin(90°) = (1)(1)(1) = 1.
- Direction: Perpendicular to both i (x-axis) and j (y-axis). Using the right-hand rule (curl fingers from i to j), your thumb points along the positive z-axis.
- Therefore, i × j = k.
-
Vector j × Vector i:
- Magnitude: |j × i| = |j| |i| sin(90°) = (1)(1)(1) = 1.
- Direction: Perpendicular to both j (y-axis) and i (x-axis). Using the right-hand rule (curl fingers from j to i), your thumb points along the negative z-axis.
- Therefore, j × i = -k.
This example demonstrates both the magnitude calculation and how the direction changes when the order of the vectors is reversed, illustrating the anti-commutative property.
Comparison: Scalar vs. Vector Product
Here's a quick comparison:
| Feature | Scalar Product (Dot Product) | Vector Product (Cross Product) |
|---|---|---|
| Notation | a ⋅ b | a × b |
| Result | A scalar (a single number) | A vector |
| Magnitude | a | |
| Direction | None (it's a scalar) | Perpendicular to both vectors (a and b) |
| Commutativity | Commutative (a ⋅ b = b ⋅ a) | Anti-commutative (a × b = -b × a) |
Understanding the vector product is fundamental for analyzing phenomena involving rotation, magnetic fields, and other areas where direction perpendicular to a plane is crucial.
