Cross Product Calculator: 3D Vectors With Steps
This cross product calculator multiplies two vectors in three dimensions. Enter vector a and vector b as three numbers each, like 1, 2, 3 and 4, 5, 6 (2D vectors are treated as having z = 0). You get a × b in component form and as i, j, k notation, with the determinant steps for each component. The calculator also gives the magnitude |a × b|, which equals the area of the parallelogram the vectors span, half of it for the triangle, the unit normal vector, the angle between a and b, and checks that the result is perpendicular to both.
Cross Product Calculator
Find the cross product a × b of two 3D vectors, with the determinant steps, magnitude and angle.
How to use the Cross Product Calculator
- Enter the three components of vector a.
- Enter the three components of vector b.
- Read a × b and the determinant steps.
- Use the magnitude for the parallelogram or triangle area.
Cross product formula
a × b = (a₂b₃ − a₃b₂) i + (a₃b₁ − a₁b₃) j + (a₁b₂ − a₂b₁) k · |a × b| = |a||b| sin ฮธ
Features
- a × b with i, j, k steps
- Magnitude and unit normal
- Parallelogram and triangle area
- Angle between vectors
- Perpendicular check with dot products
Frequently asked questions
How do you calculate a cross product?
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1, 2, 3) × (4, 5, 6) that is (12 − 15, 12 − 6, 5 − 8) = (−3, 6, −3).
What does the cross product represent?
A vector perpendicular to both a and b whose length equals the area of the parallelogram formed by a and b. Its direction follows the right-hand rule.
Is the cross product commutative?
No. b × a = −(a × b), so swapping the order flips the direction.
What is the cross product of parallel vectors?
The zero vector, because the sine of 0° or 180° is 0.
What is the difference between dot and cross product?
The dot product gives a number (a scalar) related to the cosine of the angle. The cross product gives a vector related to the sine of the angle.