Cross product
Cross Product
The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to the plane containing the two input vectors. The cross product is widely used in physics and engineering to find a vector that is perpendicular to two given vectors.
Definition
Given two vectors A and B, their cross product C = A × B is defined as:
C = A × B = |A| |B| sin(θ) n
where:
- |A| and |B| are the magnitudes of vectors A and B respectively.
- θ is the angle between A and B (0 ≤ θ ≤ π).
- n is a unit vector perpendicular to the plane containing A and B, following the right-hand rule.
Formula
If vectors A and B are represented in Cartesian coordinates as:
A = (A_x, A_y, A_z) B = (B_x, B_y, B_z)
Then their cross product C = A × B is given by:
C = (C_x, C_y, C_z) = (A_yB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y - A_yB_x)
Properties
The cross product has several important properties:
| Property | Description |
|---|---|
| Anticommutative | A × B = -(B × A) |
| Distributive | A × (B + C) = A × B + A × C |
| Scalar Multiplication | (k*A) × **B* = k(A × B) = A × (k*B*) |
| Perpendicular Vector | A × B is perpendicular to both A and B |
| Magnitude | \ |
Examples
Example 1: Basic Cross Product Calculation
Calculate the cross product of A = (3, -3, 1) and B = (4, 9, 2).
A × B = ((-3)(2) - (1)(9), (1)(4) - (3)(2), (3)(9) - (-3)(4)) = (-6 - 9, 4 - 6, 27 + 12) = (-15, -2, 39)
Example 2: Perpendicularity
Show that A × B is perpendicular to both A and B using the vectors from Example 1.
A · (A × B) = (3, -3, 1) · (-15, -2, 39) = 3(-15) + (-3)(-2) + 1(39) = -45 + 6 + 39 = 0
B · (A × B) = (4, 9, 2) · (-15, -2, 39) = 4(-15) + 9(-2) + 2(39) = -60 - 18 + 78 = 0
Since the dot product of A with A × B and B with A × B are both zero, this confirms that A × B is perpendicular to both A and B.
Example 3: Area of a Parallelogram
Find the area of the parallelogram formed by vectors A = (1, 0, 0) and B = (0, 1, 0).
First, calculate A × B:
A × B = (0(0) - 0(1), 0(0) - 1(0), 1(1) - 0(0)) = (0, 0, 1)
The magnitude of A × B gives the area of the parallelogram:
Area = |A × B| = √(0^2 + 0^2 + 1^2) = √1 = 1
The area of the parallelogram is 1 square unit.
Conclusion
The cross product is a fundamental operation in vector calculus, providing a way to find a vector perpendicular to a plane defined by two vectors. It has applications in physics, engineering, and computer graphics, among other fields. Understanding the properties and how to calculate the cross product is essential for solving problems involving three-dimensional vectors.