Cross Product Calculator
Two vectors in 3D space hide a third one: the vector that stands at right angles to both of them. That vector is the cross product A x B, and getting its three components right by hand is where sign errors creep in, especially on the middle component. This calculator does the component arithmetic for you and shows every step.
Enter the x, y and z components of vectors A and B below. You get A x B, its magnitude, the parallelogram and triangle areas, the unit normal, and B x A so you can watch the sign flip, plus a 3D drawing built from the exact numbers you entered.
The 3D drawing is a perspective-style projection of your actual inputs, not a generic stock diagram. For parallel inputs it collapses to a line, which is exactly why the cross product is the zero vector in that case.
Quick Answer
Key takeaways
- The cross product returns a vector, not a number. It is perpendicular to both inputs, and the right-hand rule picks which of the two perpendicular directions it takes.
- Its length means something concrete: |A x B| = |A| |B| sin(theta), the area of the parallelogram built on A and B. The triangle on the same two sides has half that area.
- Order is not a detail. Swapping the inputs negates every component, so B x A = -(A x B). The calculator prints both so the flip is visible on your own numbers.
- A zero result is information, not an error. It means the vectors are parallel, or one of them is the zero vector, and no unique perpendicular direction exists.
- Quick Answer
- How to use this cross product calculator
- The cross product formula, including the sign everyone drops
- Worked example: (1, 2, 3) crossed with (4, 5, 6)
- Where the cross product does real work
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to use this cross product calculator
Enter all six components, three for A and three for B, then press Calculate. The fields accept decimals and negatives, and every component is required: a missing component stops the calculation with a note naming the field, because a blank is not the same thing as a zero.
Read the headline vector first. A x B is printed as (x, y, z). The line under it confirms the result is perpendicular to both inputs, or tells you the inputs are parallel when the result is zero.
Scan the result grid. Magnitude, parallelogram area, triangle area, unit normal and B x A sit side by side, so you do not need a second tool for the follow-on quantities most problems ask for next.
Check the drawing and the steps. The 3D view draws A, B, the result and the shaded parallelogram from your inputs. Under it, the component working shows each multiplication and subtraction, so you can follow or audit every number.
One habit worth building: before you trust a hand calculation, run it here and compare the y component. The middle component is where most manual errors happen, for a reason the formula section explains.
The cross product formula, including the sign everyone drops
The cross product is defined component by component. Each output component is built from the two input components that do not share its axis.
The y component is the trap. Written as a determinant, the j term carries a built-in minus sign: the component is AzBx - AxBz, which is the negative of the AxBz - AzBx pattern your eye expects from the x and z rows. Textbooks such as the cross product chapter in LibreTexts Calculus 3 flag exactly this step, and it is why the steps box in the results prints the y line in the same AzBx - AxBz order the formula uses.
Direction comes from the right-hand rule: point the fingers of your right hand along A, curl them toward B, and your thumb gives the direction of A x B. That is also why the operation only exists in three dimensions in this form. The dot product, which returns a scalar built on the cosine of the angle, works in any dimension; the cross product trades that generality for a direction. If your problem is really about the angle itself, the cosine calculator covers the trig side of that comparison.
Two quick self-checks cost nothing. First, the result must be perpendicular to both inputs, so A. (A x B) and B. (A x B) both equal zero; the worked example below shows that check with real numbers. Second, crossing a vector with itself, or with any multiple of itself, must give (0, 0, 0).
Worked example: (1, 2, 3) crossed with (4, 5, 6)
The calculator is pre-filled with this pair, and every figure in the table is what it returns. Press Calculate and compare line by line.
| Step | Working | Result |
|---|---|---|
| x component | AyBz - AzBy = 2 x 6 - 3 x 5 = 12 - 15 | -3 |
| y component | AzBx - AxBz = 3 x 4 - 1 x 6 = 12 - 6 | 6 |
| z component | AxBy - AyBx = 1 x 5 - 2 x 4 = 5 - 8 | -3 |
| A x B | Combine the three components | (-3, 6, -3) |
| Magnitude | sqrt(9 + 36 + 9) = sqrt(54) | 7.348469 |
| Parallelogram area | Equal to the magnitude | 7.348469 |
| Triangle area | Half the magnitude | 3.674235 |
| Unit normal | (A x B) / 7.348469 | (-0.408248, 0.816497, -0.408248) |
| B x A | Negate every component | (3, -6, 3) |
Now the perpendicular check. A. (A x B) = 1 x (-3) + 2 x 6 + 3 x (-3) = -3 + 12 - 9 = 0. B. (A x B) = 4 x (-3) + 5 x 6 + 6 x (-3) = -12 + 30 - 18 = 0. Both dot products vanish, which confirms the result really is perpendicular to both inputs. Two benchmark cases are worth memorising as sanity checks: (1, 0, 0) x (0, 1, 0) = (0, 0, 1), the unit-vector cycle i x j = k, and any parallel pair such as (2, 2, 2) x (4, 4, 4) returns (0, 0, 0).
Where the cross product does real work
Surface normals are the everyday use. A flat surface in 3D is fixed by two directions lying in it, and their cross product is the normal: the direction the surface faces. Graphics engines compute normals this way for every polygon, because lighting, shading and visibility all depend on which way a face points. Game and modelling tools repeat this calculation millions of times; doing it once, slowly, with the steps visible is how the pattern sticks.
Physics runs on the same operation. Torque is r x F, the position vector crossed with the force, and its magnitude |r| |F| sin(theta) explains why pushing a door near the hinge barely turns it: the angle and the lever arm shrink the product. Angular velocity works the same way in reverse, with v = omega x r giving the velocity of a point on a rotating body. If you then need how that motion changes, the acceleration calculator picks up the kinematics from there.
Geometry problems use the outputs printed next to the main result. The parallelogram and triangle areas come straight from the magnitude, and the same trick builds the volume of a parallelepiped when a third vector joins in (the scalar triple product). Mesh software leans on those face areas when it works out quantities such as the centroid calculator handles for composite shapes. In linear algebra proper, the cross product reappears as the null-space direction of a 2 x 3 system, a cousin of the eigenvector work in the complex eigenvector calculator.
What competing calculators miss
The tools ranking for this keyword split into two camps: bare answer boxes and everything-at-once vector suites. Both leave gaps this page was built to close.
Miss 1: the result with no working. Several top results return the vector and its magnitude and stop. When your homework or your code disagrees with the answer, there is nothing to compare against. Here the steps box prints each component as a separate line in the formula's own order, including the awkward y line, so the exact step that diverged is easy to spot.
Miss 2: the follow-on quantities live elsewhere. Most problems that ask for A x B ask for the unit normal, an area, or both in the next breath. Suite-style tools bury those inside a six-operation panel; bare tools omit them. This page prints magnitude, parallelogram area, triangle area, unit normal and B x A in one grid, because they all come from the same three components.
Miss 3: a generic diagram, or none. A static picture of two arbitrary vectors does not tell you whether your result points the right way. The drawing here is rendered from your components: your A in blue, your B in amber, your A x B in red, and the shaded parallelogram whose area is the magnitude.
Miss 4: the zero case treated as a non-answer. Tools that print (0, 0, 0) with no comment leave users wondering if they mistyped. This calculator says plainly that the vectors are parallel, and marks the unit normal as undefined rather than inventing a direction.
One honest limit: this is a 3D tool. Two-dimensional work is handled by entering z = 0 for both vectors (the answer then points along z), and the seven-dimensional cross product from advanced algebra is out of scope here, as it is for every mainstream calculator.
Frequently Asked Questions
Finding a vector perpendicular to two given vectors, such as a surface normal in 3D graphics, computing the areas of parallelograms and triangles in space, and physics quantities defined as cross products, including torque (r x F) and angular momentum.
Yes. Enter 0 for Az and Bz. The x and y components of the result come out as 0, and the z component is AxBy - AyBx, the 2D determinant, which also gives the signed area of the parallelogram the two 2D vectors span.
The cross product is anticommutative. Swapping the inputs reverses the curl direction in the right-hand rule, so the result points the opposite way while keeping the same magnitude. The calculator prints B x A next to A x B so you can see every component flip sign.
The two vectors are parallel, meaning one is a scalar multiple of the other, or at least one of them is the zero vector. The parallelogram they span has zero area, and there is no unique perpendicular direction, so the unit normal is undefined.
The unit normal is A x B divided by its own magnitude, a vector of length 1 pointing in the cross product direction. Use it when a problem asks for direction only: a plane normal, a surface orientation in graphics, or the unit vector orthogonal to two given vectors.
Dot the result with each input. Both dot products must equal zero, because the cross product is perpendicular to both vectors. For A = (1, 2, 3) and B = (4, 5, 6), A x B = (-3, 6, -3), and both checks give 0: -3 + 12 - 9 and -12 + 30 - 18.
The dot product returns a scalar, |A| |B| cos(theta), and measures how much the vectors align. The cross product returns a vector, with magnitude |A| |B| sin(theta), and measures the area they span while pointing perpendicular to both. The dot product works in any dimension; the cross product in this form is a 3D operation.
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References
- LibreTexts, Calculus 3 (SUNY Geneseo), "The Cross Product": component formula, determinant form, right-hand rule, and the unit-vector cycle used for the benchmark checks. LibreTexts: The Cross Product (Calculus 3). Accessed October 2026.
- LibreTexts, Interactive Calculus, "The Cross Product": magnitude as parallelogram area and the scalar triple product for parallelepiped volume. LibreTexts: The Cross Product (Interactive Calculus). Accessed October 2026.
- Khan Academy, Linear Algebra, "Cross product introduction": the definition of A x B, why it is defined only in three dimensions, and how it differs from the dot product. Khan Academy: Cross product introduction. Accessed October 2026.
- Khan Academy, Linear Algebra, "Proof: relationship between cross product and sin of angle": the proof that |A x B| = |A| |B| sin(theta). Khan Academy: Cross product and sin of angle proof. Accessed October 2026.
- LibreTexts, Further Applications of Trigonometry, "The Cross Product": orthogonality proofs showing the result is perpendicular to both inputs, the basis of the dot-product self-check. LibreTexts: The Cross Product (Trigonometry). Accessed October 2026.
- Paul Dawkins, Paul's Online Math Notes, Calculus II, "Cross Product": worked practice problems with full solutions for the determinant method. Paul's Online Math Notes: Cross Product problem and solution. Accessed October 2026.