Vector product
Vector Product
Vector product, also known as the cross product, is a binary operation on two vectors in three-dimensional space. It is denoted by the symbol ×. The vector product of two vectors results in a vector that is perpendicular to the plane containing the original vectors.
Definition
Given two vectors a and b, their vector product c is defined as:
c = a × b
The magnitude of c is given by the area of the parallelogram with sides a and b:
|c| = |a| |b| sin(θ)
where θ is the angle between a and b (0 ≤ θ ≤ π).
The direction of c is given by the right-hand rule: if the fingers of your right hand point from a to b, then your thumb points in the direction of c.
Formula
In terms of components, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then the vector product is:
c = a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
Properties
The vector product has several important properties:
- Anticommutativity: a × b = -(b × a)
- Distributivity: a × (b + c) = a × b + a × c
- Scalar Multiplication: (k*a) × **b* = k(a × b) = a × (k*b*), where k is a scalar
- Orthogonality: a × b is orthogonal to both a and b
- Zero Vector: If a × b = 0, then a and b are parallel (or one of them is the zero vector)
Differences and Important Points
| Property | Dot Product (Scalar Product) | Cross Product (Vector Product) |
|---|---|---|
| Symbol | a · b | a × b |
| Result | Scalar | Vector |
| Direction | Not applicable | Perpendicular to a and b |
| Magnitude | a | |
| Commutativity | Commutative | Anticommutative |
| Zero Result | If a and b are orthogonal | If a and b are parallel or one is zero |
| Application | Projection, Work | Torque, Angular momentum |
Examples
Example 1: Compute the Cross Product
Given two vectors a = (3, -3, 1) and b = (4, 9, 2), find the vector product a × b.
a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁) = ((-3)×2 - 1×9, 1×4 - 3×2, 3×9 - (-3)×4) = (-6 - 9, 4 - 6, 27 + 12) = (-15, -2, 39)
So, a × b = (-15, -2, 39).
Example 2: Verify Anticommutativity
Using the vectors from Example 1, verify that a × b = -(b × a).
b × a = (9×1 - 2×(-3), 2×3 - 4×1, 4×(-3) - 9×3) = (9 + 6, 6 - 4, -12 - 27) = (15, 2, -39)
Now, -(b × a) = -1 × (15, 2, -39) = (-15, -2, 39), which is equal to a × b.
Example 3: Orthogonality
Show that a × b is orthogonal to a and b.
Using the vectors from Example 1, a × b = (-15, -2, 39).
Now, compute the dot product of a × b with a and b:
(a × b) · a = (-15, -2, 39) · (3, -3, 1) = -15×3 - 2×(-3) + 39×1 = -45 + 6 + 39 = 0
(a × b) · b = (-15, -2, 39) · (4, 9, 2) = -15×4 - 2×9 + 39×2 = -60 - 18 + 78 = 0
Since the dot products are zero, a × b is orthogonal to both a and b.
The vector product is a fundamental operation in vector calculus and is widely used in physics and engineering to solve problems involving rotational forces, magnetic fields, and more. Understanding the cross product is essential for anyone studying vector calculus or working in related fields.