Cross Product Calculator

Definition:

The cross product of two vectors A and B is denoted as A × B. Its magnitude and direction are defined as:

A × B = |A| |B| sin(θ) n

where:

• |A| and |B| are the magnitudes (lengths) of vector A and vector B, respectively.
• θ is the smallest angle between vector A and vector B (0° ≤ θ ≤ 180°).
• n is the unit vector perpendicular to both A and B, following the right-hand rule.

Component Form:

When vectors A and B are given in their Cartesian components (A = Ax i + Ay j + Az k and B = Bx i + By j + Bz k), their cross product can be calculated as a determinant:

A × B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k

This formula provides the x, y, and z components of the resulting vector.

Key Properties:

• Anti-commutative: The order of vectors matters. Swapping the order reverses the direction of the resulting vector: A × B = -(B × A).
• Distributive: The cross product distributes over vector addition: A × (B + C) = (A × B) + (A × C).
• Not associative: The grouping of vectors in multiple cross products changes the result: (A × B) × C ≠ A × (B × C).
• Scalar multiplication: A scalar multiplier can be applied before or after the cross product: (kA) × B = k(A × B) = A × (kB).

Special Cases:

• Parallel vectors: If two vectors A and B are parallel or collinear (θ = 0° or 180°), their cross product is the zero vector: A × B = 0. This is because sin(0°) = sin(180°) = 0.
• Perpendicular vectors: If two vectors A and B are perpendicular (orthogonal, θ = 90°), the magnitude of their cross product is simply the product of their magnitudes: |A × B| = |A| |B|. This is because sin(90°) = 1.
• Unit vectors: The cross product of standard unit vectors (i, j, k) follows a cyclic pattern: i × j = k, j × k = i, k × i = j. Also, i × i = j × j = k × k = 0.

Area Calculations

The magnitude of the cross product of two vectors A and B represents the area of the parallelogram formed by these two vectors when they share a common origin.

Area of parallelogram = |A × B|

Consequently, the area of a triangle formed by two vectors is half the magnitude of their cross product.

Area of triangle = ½|A × B|

Normal Vector

The cross product provides a vector that is perpendicular (normal) to the plane containing the two original vectors. This is incredibly useful for defining the orientation of surfaces in 3D geometry.

Unit normal = (A × B)/|A × B|

Torque in Physics

In physics, torque (τ) is a rotational force that causes an object to rotate about an axis. It is defined as the cross product of the position vector (r) from the pivot point to the point where the force is applied, and the force vector (F).

τ = r × F

Angular Momentum

Angular momentum (L) is a measure of the rotational inertia of a rotating object. For a particle, it is defined as the cross product of its position vector (r) relative to the origin and its linear momentum vector (p).

L = r × p