Cross Product Calculator
A cross product calculator finds the cross product of two three-dimensional vectors. The cross product produces a new vector that is perpendicular to both original vectors and is commonly used in vector mathematics, physics, engineering, and geometry.
What Is a Cross Product?
The cross product, also called the vector product, is an operation between two vectors in three-dimensional space. For vectors A and B, the result is written as:
A Γ B
The resulting vector is perpendicular to both A and B.
Cross Product Formula
If:
A = (a1, a2, a3)
B = (b1, b2, b3)
then:
A Γ B = (a2b3 β a3b2, a3b1 β a1b3, a1b2 β a2b1)
How to Calculate a Cross Product
Consider the vectors:
A = (1, 2, 3)
B = (4, 5, 6)
Calculate each component:
x = (2 Γ 6) β (3 Γ 5) = β3
y = (3 Γ 4) β (1 Γ 6) = 6
z = (1 Γ 5) β (2 Γ 4) = β3
Therefore:
A Γ B = (β3, 6, β3)
Cross Product Using a Determinant
The cross product can also be represented using a determinant:
A Γ B = | i j k |
| a1 a2 a3 |
| b1 b2 b3 |
Expanding this determinant gives the component formula used above.
Magnitude of the Cross Product
The magnitude of the cross product is:
|A Γ B| = |A||B|sin(ΞΈ)
Here, ΞΈ is the angle between the two vectors. The magnitude is also equal to the area of the parallelogram formed by the two vectors.
Cross Product vs. Dot Product
The cross product and dot product are different vector operations. A cross product produces a vector, while a dot product produces a scalar value.
| Operation | Result | Main Property |
|---|---|---|
| Cross product | Vector | Perpendicular to both vectors |
| Dot product | Scalar | Related to the angle between vectors |
Important Properties
The cross product is not commutative. Reversing the order changes the direction of the result:
A Γ B = β(B Γ A)
If two vectors are parallel, their cross product is the zero vector because the angle between them is 0Β° or 180Β°.
Where Is the Cross Product Used?
Cross products are used in physics to calculate torque and angular momentum, as well as in geometry, computer graphics, engineering, robotics, and three-dimensional vector calculations.
Common Mistakes to Avoid
Keep the vector components in the correct order when applying the formula. The middle component is particularly easy to get wrong because of its subtraction order.
Also remember that changing the order of the two vectors reverses the direction of the result.
Quick Summary
The cross product of two 3D vectors produces a vector perpendicular to both. For A = (a1, a2, a3) and B = (b1, b2, b3), calculate each component using the cross product formula and combine them into the resulting vector.