Cross Product Calculator
Calculate the cross product of two 3D vectors with the determinant method, step by step, plus its magnitude, the angle between the vectors, areas and a 3D view.
Try an example
Cross product · resulting vector
Perpendicular to A and B. A × B is perpendicular to both A and B. Its direction follows the right-hand rule and its length is the area of the parallelogram spanned by A and B.
Each component from a 2 × 2 determinant
x-component (i)
a₂b₃ − a₃b₂
(2 × 6) − (3 × 5) = 12 − 15
x = −3
y-component (j)
a₃b₁ − a₁b₃
(3 × 4) − (1 × 6) = 12 − 6
y = 6
z-component (k)
a₁b₂ − a₂b₁
(1 × 5) − (2 × 4) = 5 − 8
z = −3
Magnitude, angle and area
- Magnitude |A × B|
- 3√6 ≈ 7.3485
- |A| and |B|
- √14 ≈ 3.7417 and √77 ≈ 8.775
- sin θ = |A × B| / (|A||B|)
- ≈ 0.223814
- Angle θ between A and B (degrees)
- ≈ 12.9332° (≈ 0.2257 rad)
- Parallelogram area (geometric interpretation)
- 3√6 ≈ 7.3485 square units
- Triangle area = ½|A × B|
- 3√6/2 ≈ 3.6742 square units
Perpendicularity check
Perpendicular to A: A · (A × B) = 0
Perpendicular to B: B · (A × B) = 0
A dot product of 0 means the vectors are at right angles, so A × B is perpendicular to both A and B.
Order matters: B × A
B × A = (3, −6, 3)
= −(A × B)
The cross product is anticommutative: swapping the vectors reverses the direction but keeps the length. Try the Swap A and B button.
3D view
Right-hand rule: point the fingers of your right hand along A and curl them toward B; your thumb points along A × B. For A along x and B along y, the thumb points up the z-axis: i × j = k.
Step-by-step determinant calculation
Step 1 — Set up the determinant
Put the unit vectors i, j, k in the first row, A in the second row and B in the third.
| i j k |
A × B = | 1 2 3 |
| 4 5 6 |
Step 2 — Expand along the first row
Each unit vector is multiplied by the 2 × 2 determinant left after crossing out its row and column. The j term takes a minus sign.
A × B = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
= (12 − 15)i − (6 − 12)j + (5 − 8)k
= (−3)i − (−6)j + (−3)k
= −3i + 6j − 3k
Step 3 — Write each component
The minus sign in front of j is absorbed by reversing the order: y = a₃b₁ − a₁b₃.
x = a₂b₃ − a₃b₂: (2 × 6) − (3 × 5) = 12 − 15 = −3
y = a₃b₁ − a₁b₃: (3 × 4) − (1 × 6) = 12 − 6 = 6
z = a₁b₂ − a₂b₁: (1 × 5) − (2 × 4) = 5 − 8 = −3
A × B = (−3, 6, −3)
Step 4 — Magnitude of the cross product
C = A × B = (−3, 6, −3)
|C| = √(c₁² + c₂² + c₃²)
= √((−3)² + 6² + (−3)²)
= √(9 + 36 + 9) = √54
= 3√6 ≈ 7.3485
Step 5 — Angle between A and B
From |A × B| = |A||B| sin θ. sin⁻¹ only returns angles from 0° to 90°, so the sign of A · B = 32 decides whether θ is acute or obtuse.
|A| = √14 ≈ 3.7417, |B| = √77 ≈ 8.775
sin θ = |A × B| / (|A||B|) = 3√6 / (√14 × √77) ≈ 0.223814
sin⁻¹(0.223814) ≈ 12.9332°
A · B ≥ 0, so θ ≈ 12.9332°
Step 6 — Check: A × B is perpendicular to A and to B
The dot product of perpendicular vectors is 0.
A · (A × B) = (1)(−3) + (2)(6) + (3)(−3) = 0
B · (A × B) = (4)(−3) + (5)(6) + (6)(−3) = 0
Cross product formula
Determinant form
| i j k |
A × B = | a₁ a₂ a₃ |
| b₁ b₂ b₃ |Component form
A × B = (a₂b₃ − a₃b₂,
a₃b₁ − a₁b₃,
a₁b₂ − a₂b₁)
Magnitude
|A × B| = |A||B| sin θ
Areas
Parallelogram = |A × B|, triangle = ½|A × B|
Cross product properties
| Anticommutative | A × B = −(B × A) | Swapping the order reverses the direction. |
|---|---|---|
| Distributive | A × (B + C) = A × B + A × C | Works over vector addition. |
| Scalar multiples | (kA) × B = k(A × B) = A × (kB) | Scaling a factor scales the result. |
| Self cross product | A × A = 0 | Any vector is parallel to itself. |
| Perpendicular | A · (A × B) = B · (A × B) = 0 | The result is at right angles to both vectors. |
| Unit vectors | i × j = k, j × k = i, k × i = j | Reverse any pair and the sign flips: j × i = −k. |
Cross product vs dot product
| Cross product A × B | Dot product A · B | |
|---|---|---|
| Result | A vector | A scalar (a number) |
| Written | A × B | A · B |
| Size uses | |A||B| sin θ | |A||B| cos θ |
| Zero when | A and B are parallel (or one is zero) | A and B are perpendicular (or one is zero) |
| Order | A × B = −(B × A) | A · B = B · A |
| Useful for | Perpendicular directions, area, torque, surface normals | Angles, projections, testing perpendicularity |
To find the angle between vectors or test whether they are perpendicular, use the Dot Product Calculator. The Vector Calculator handles addition, magnitude and unit vectors in 2D and 3D. The triangle area ½|A × B| can be cross-checked with the Triangle Calculator, and sin⁻¹ and other trigonometry is available in the Scientific Calculator.
This cross product calculator finds A × B for two 3D vectors. Enter the x, y and z components of Vector A and Vector B as the rows of a determinant, and the calculator shows the resulting vector, for example (1, 2, 3) × (4, 5, 6) = (−3, 6, −3), with the determinant expansion and every component worked out.
It also gives the magnitude |A × B|, the angle between A and B from |A × B| = |A||B| sin θ, the areas of the parallelogram and triangle formed by the vectors, and a check that the result is perpendicular to both inputs. A rotatable 3D view shows A × B standing at right angles to the plane of A and B, and parallel vectors, antiparallel vectors and zero vectors are detected and explained.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Standard: (1, 2, 3) × (4, 5, 6) | (−3, 6, −3) | x = 2·6 − 3·5 = 12 − 15 = −3. y = 3·4 − 1·6 = 12 − 6 = 6. z = 1·5 − 2·4 = 5 − 8 = −3. Check: (1, 2, 3) · (−3, 6, −3) = −3 + 12 − 9 = 0. |A × B| = √54 = 3√6 ≈ 7.35. |
| Parallel vectors: (1, 2, 3) × (2, 4, 6) | (0, 0, 0) | x = 2·6 − 3·4 = 0. y = 3·2 − 1·6 = 0. z = 1·4 − 2·2 = 0. B = 2A, so the vectors are parallel and span no area. |
| Unit vectors: (1, 0, 0) × (0, 1, 0) | (0, 0, 1) | x = 0·0 − 0·1 = 0. y = 0·0 − 1·0 = 0. z = 1·1 − 0·0 = 1. So i × j = k: the result points along the z-axis, perpendicular to both the x- and y-axes. |
| Reverse order: (4, 5, 6) × (1, 2, 3) | (3, −6, 3) | x = 5·3 − 6·2 = 3. y = 6·1 − 4·3 = −6. z = 4·2 − 5·1 = 3. This is −(−3, 6, −3): swapping the vectors reverses the direction. |
What Is the Cross Product?
The cross product A × B combines two 3D vectors into a third vector that is perpendicular to both of them. Unlike the dot product A · B, which produces a single number, the cross product produces a vector with three components.
For the unit vectors along the axes, i × j = k, j × k = i and k × i = j. Its length measures how far from parallel the two vectors are: it is largest when they are perpendicular and zero when they are parallel.
The Determinant Method
Write i, j and k in the first row of a 3 × 3 determinant, the components of A in the second row and those of B in the third. Expand along the first row: A × B = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k.
The minus sign in front of the j term is the most common source of mistakes. Absorbing it gives the component form A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1, 2, 3) × (4, 5, 6): x = 2·6 − 3·5 = −3, y = 3·4 − 1·6 = 6, z = 1·5 − 2·4 = −3.
Geometric Meaning: Perpendicular Direction and Area
A × B is perpendicular to the plane containing A and B, which you can confirm with dot products: A · (A × B) = 0 and B · (A × B) = 0. Its direction follows the right-hand rule: point the fingers of your right hand along A, curl them toward B, and your thumb points along A × B.
Its magnitude is |A × B| = |A||B| sin θ, where θ is the angle between A and B. That is exactly the area of the parallelogram with sides A and B, so the triangle with sides A and B has area ½|A × B|. For A = (3, 0, 0) and B = (1, 2, 0), A × B = (0, 0, 6): the parallelogram has area 6 square units and the triangle 3.
Parallel Vectors and the Order of Multiplication
If A and B are nonzero and A × B = (0, 0, 0), the vectors are parallel (θ = 0°) or antiparallel (θ = 180°), because sin θ = 0. For example, (1, 2, 3) × (2, 4, 6) = (0, 0, 0). If one vector is the zero vector, the cross product is also zero, but only because the zero vector has no length or direction.
The cross product is anticommutative: B × A = −(A × B). Swapping the vectors reverses the direction of the result without changing its length, so (4, 5, 6) × (1, 2, 3) = (3, −6, 3).
Where the Cross Product Is Used
In physics, torque is τ = r × F, the cross product of the position vector from the pivot and the applied force; angular momentum is L = r × p. In electromagnetism, the magnetic force on a moving charge is F = qv × B, perpendicular to both the velocity and the magnetic field.
In 3D computer graphics and geometry, the cross product of two edges of a triangle gives the surface normal used for lighting and for deciding which way a face points, and half its length gives the triangle's area. Robotics and engineering use it for rotations, moments and finding directions perpendicular to two given ones.
How to Use the Cross Product Calculator
- Enter the x, y and z components of Vector A (x₁, y₁, z₁) in the second row of the determinant.
- Enter the x, y and z components of Vector B (x₂, y₂, z₂) in the third row. The result updates as you type, with no Calculate button needed.
- Read the resulting vector A × B in component form and in i, j, k notation.
- Follow the determinant expansion and the component-by-component calculation, including the minus sign on the j term.
- Check the magnitude |A × B|, the angle between A and B, the parallelogram and triangle areas, and the perpendicularity check.
- Drag the 3D view, or use the Rotate and Tilt sliders, to see A × B standing perpendicular to the shaded plane of A and B. Use Swap A and B to see the sign reverse.
Frequently Asked Questions
What is a cross product?
An operation on two 3D vectors that returns a third vector perpendicular to both. Its length is |A||B| sin θ and its direction follows the right-hand rule. For example, i × j = k.
How do you calculate the cross product?
Set up the determinant with i, j, k in the first row, A in the second and B in the third, and expand: A × B = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k. For (1, 2, 3) × (4, 5, 6) this gives (−3, 6, −3).
What is the cross product formula?
A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Each component uses the other two coordinates, and the middle one is a₃b₁ − a₁b₃ (not a₁b₃ − a₃b₁).
Why is the cross product perpendicular to both vectors?
Because its dot product with each of them is zero. Expanding A · (A × B) gives a₁(a₂b₃ − a₃b₂) + a₂(a₃b₁ − a₁b₃) + a₃(a₁b₂ − a₂b₁), and every term cancels. The same happens for B.
What does a zero cross product mean?
Either the two vectors are parallel or antiparallel (sin θ = 0), or at least one of them is the zero vector. The parallelogram they span has no area.
Are parallel vectors’ cross products zero?
Yes. If B = kA for some number k, then A × B = k(A × A) = 0. For example, (1, 2, 3) × (2, 4, 6) = (0, 0, 0), and so does any vector crossed with itself.
What is the difference between dot product and cross product?
The dot product A · B is a scalar equal to |A||B| cos θ, used for angles and projections. The cross product A × B is a vector of length |A||B| sin θ, perpendicular to both, used for normals, areas and torque.
How do you find the magnitude of a cross product?
Take the length of the resulting vector: |A × B| = √(x² + y² + z²). For (−3, 6, −3), |A × B| = √(9 + 36 + 9) = √54 = 3√6 ≈ 7.3485. It also equals |A||B| sin θ.
How can the cross product be used to find area?
The parallelogram with sides A and B has area |A × B|, and the triangle with sides A and B has area ½|A × B|. For a triangle with vertices P, Q, R, use A = Q − P and B = R − P.
What is the right-hand rule?
A way to find the direction of A × B: point the fingers of your right hand along A and curl them toward B; your thumb points along A × B. Reversing the order makes the thumb point the opposite way.
Is the cross product defined for 2D vectors?
Not as a vector in the plane. The usual approach is to treat 2D vectors as 3D vectors with z = 0: (a₁, a₂, 0) × (b₁, b₂, 0) = (0, 0, a₁b₂ − a₂b₁), which points along the z-axis. The number a₁b₂ − a₂b₁ is sometimes called the 2D cross product.
Why does changing the order of the vectors change the result?
The cross product is anticommutative: B × A = −(A × B). Each component is a difference such as a₂b₃ − a₃b₂, and swapping A and B swaps the two terms, which flips every sign. The length stays the same.
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Last updated: September 27, 2026.