✖️ Cross Product
Calculate the cross product (vector product) of two 3D vectors.
The Comprehensive Guide to Cross Product Calculator
What is a Cross Product Calculator?
The Cross Product Calculator computes the exact orthogonal vector produced when two 3D vectors are multiplied together. Unlike the dot product (which returns a simple scalar number), the cross product fundamentally creates a brand new third vector that is perfectly perpendicular (at a 90-degree angle) to both original input vectors.
The Mathematical Formula
This tool utilize standardized mathematical formulas and logic to calculate precise Cross Product results.
Calculation Example
Let's cross Vector u (1, -7, 1) with Vector v (5, 2, 4):
- i component (x): (-7 × 4) - (1 × 2) = -28 - 2 = -30
- j component (y): (1 × 4) - (1 × 5) = 4 - 5 = -1. Because of the formula's strictly alternating signs, we flip it: 1
- k component (z): (1 × 2) - (-7 × 5) = 2 - (-35) = 37
- Resultant Vector (w): (-30, 1, 37)
Strategic Use Cases
- Classical Mechanics (Torque): Calculating rotational force (torque). Torque is the cross product of the lever arm vector (distance from pivot) and the applied force vector.
- Electromagnetism: Determining the magnetic force exerted on a highly charged particle rapidly moving through a magnetic field (the Lorentz force).
- Computer Graphics: Finding the "normal vector" of a 3D polygon/triangle. This is absolutely critical for calculating how light realistically bounces off a 3D model during rendering.
Frequently Asked Questions
Is u × v the exact same as v × u?
No, absolutely not. The cross product is strictly 'anti-commutative'. u × v = -(v × u). Swapping the strict order of the vectors will produce a result with the exact same magnitude, but pointing in the exact opposite mathematical direction.
What does it mean if the cross product is strictly (0,0,0)?
If the cross product is a perfect zero vector, it mathematically proves that your two original vectors are either perfectly parallel to each other, perfectly anti-parallel, or at least one of them is the zero vector.
How do I calculate the cross product of 2D vectors?
You technically cannot. The cross product is a strictly 3D mathematical operation. To 'cross' 2D vectors, you must artificially add a '0' as their third (z/k) component, which will always result in a vector pointing straight up or straight down perfectly along the z-axis.