Cross Product Calculator
Cross product A × B of two 3D or 2D vectors with the i, j, k working, the magnitude (parallelogram area), the dot product and a picture. How it works
Use the Cross Product Calculator
About the Cross Product Calculator
(1, 2, 3) × (4, 5, 6) = (−3, 6, −3), a vector perpendicular to both, with magnitude ≈ 7.348469228. This cross product calculator takes two 3D or 2D vectors and shows the i, j and k working, the magnitude, which is the area of the parallelogram the vectors span, the dot product and a picture of the vectors.
Worked cross products
| A | B | A × B | |A × B| | A · B |
|---|---|---|---|---|
| (1, 2, 3) | (4, 5, 6) | (-3, 6, -3) | ≈ 7.348469228 | 32 |
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 | 0 |
| (0, 1, 0) | (1, 0, 0) | (0, 0, -1) | 1 | 0 |
| (3, -3, 1) | (4, 9, 2) | (-15, -2, 39) | ≈ 41.83300133 | -13 |
| (2, 3, 4) | (5, 6, 7) | (-3, 6, -3) | ≈ 7.348469228 | 56 |
| (2, 0, 0) | (0, 3, 0) | (0, 0, 6) | 6 | 0 |
| (0.5, 1, 0) | (0, 2, 1) | (1, -0.5, 1) | 1.5 | 2 |
| (1, 2, 3) | (2, 4, 6) | (0, 0, 0) | 0 | 28 |
The last row is a pair of parallel vectors, so the cross product is the zero vector.
Frequently asked questions
- What is the cross product formula?
- A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (3, -3, 1) × (4, 9, 2): (-3×2 − 1×9, 1×4 − 3×2, 3×9 − (-3)×4) = (-15, -2, 39).
- How do you find the cross product with the determinant method?
- Write i, j, k in the first row of a 3 × 3 determinant and the two vectors below, then expand: the i term is a₂b₃ − a₃b₂, the j term is minus (a₁b₃ − a₃b₁), and the k term is a₁b₂ − a₂b₁. The tool prints the three lines with your numbers and writes the j line with the minus already applied, as a₃b₁ − a₁b₃.
- Is A × B the same as B × A?
- No, they point in opposite directions: B × A = −(A × B). (1, 0, 0) × (0, 1, 0) = (0, 0, 1), while (0, 1, 0) × (1, 0, 0) = (0, 0, -1).
- What does a zero cross product mean?
- The vectors are parallel, or one of them is the zero vector. (1, 2, 3) × (2, 4, 6) = (0, 0, 0) because the second vector is twice the first. The tool says so, since there is then no single perpendicular direction.
- How do you find the area of a parallelogram or triangle from two vectors?
- The parallelogram’s area is |A × B| and the triangle’s is half of it. For (2, 0, 0) and (0, 3, 0), |A × B| = 6, so the triangle has area 3.
- What is the cross product of two 2D vectors?
- A number, the k component of A × B with both z values 0: k = x₁y₂ − y₁x₂. Pick 2D vectors (x, y): (1, 2) and (4, 5) give 1×5 − 2×4 = -3. A negative k means turning from A to B the short way is clockwise, so A × B points into the page; |k| = 3 is the area of the parallelogram, and the triangle has half of it.
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