๐งญ Cross Product Calculator: Vector Cross Product of Two 3D Vectors
By Shihab Mia ยท Updated 2026-08-03
Enter the components of two 3D vectors to find their cross product, its magnitude, and their dot product.
This cross product calculator finds a x b for two 3D vectors a=(a1, a2, a3) and b=(b1, b2, b3), returning the resultant vector, its magnitude, and the dot product in one step. Enter the six components and the tool shows the full working, from each of the three cross product components to whether the vectors are parallel or perpendicular. It is built for students checking linear algebra or physics homework, and for anyone who needs a fast, correct answer for torque, normal vectors, or 3D geometry.
What is the Cross Product Calculator?
The cross product is an operation on two vectors in three-dimensional space that produces a third vector, not a number. Given a=(a1, a2, a3) and b=(b1, b2, b3), the cross product a x b is defined component by component as (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1). Each component follows the same pattern: skip the matching index, multiply the other two pairs diagonally, and subtract. This is easy to get backwards on the middle term, since it is written a3b1 - a1b3 rather than a1b3 - a3b1, so it is worth double-checking that term specifically when working by hand.
What makes the cross product different from the dot product is the result itself. The dot product a.b = a1b1 + a2b2 + a3*b3 collapses two vectors into a single scalar that tells you about the angle between them, while the cross product a x b produces a brand new vector that points in a direction perpendicular to both a and b, following the right-hand rule. If you point your right hand's fingers along a and curl them toward b, your thumb points along a x b. This perpendicularity is the whole reason the cross product exists: it is the standard way to generate a vector that is orthogonal to a plane defined by two other vectors, which comes up constantly in physics, engineering, and 3D graphics.
The magnitude of the cross product, |a x b| = sqrt(c1^2 + c2^2 + c3^2) where (c1, c2, c3) is the resultant vector, has its own geometric meaning: it equals the area of the parallelogram formed by a and b, and it also equals |a| |b| sin(theta), where theta is the angle between the two original vectors. This means the magnitude is largest when the vectors are perpendicular (sin 90 degrees = 1) and shrinks toward zero as the vectors become more parallel, since sin of a small angle approaches zero. That is why a zero cross product is the direct test for parallel vectors, and it is also why the dot product being zero is the direct test for perpendicular vectors, the two relationships this calculator flags automatically.
A few properties are worth internalizing before you rely on the cross product in a proof or a physics problem. It is anti-commutative, meaning b x a = -(a x b), so swapping the order flips the sign of every component rather than leaving the answer unchanged. It is not associative in the way multiplication is: (a x b) x c generally does not equal a x (b x c). And the cross product of any vector with itself, or with a parallel vector, is always the zero vector, because sin(0) = 0. These rules explain most of the surprising results people run into when they first learn the operation, and they are the same rules this calculator applies under the hood.
When to use it
- Finding a vector normal to a plane defined by two other vectors, needed for lighting and surface calculations in 3D graphics and CAD.
- Computing torque in physics, since torque equals the cross product of the position vector and the force vector, tau = r x F.
- Checking whether two 3D vectors are parallel or perpendicular quickly, without drawing a diagram or computing an angle first.
- Verifying linear algebra or vector calculus homework where the cross product, magnitude, and dot product are asked for together.
- Calculating the area of a parallelogram or a triangle in 3D space from two edge vectors, using the magnitude of their cross product.
- Determining angular momentum and magnetic force problems in physics, both of which are defined using the cross product.
How to use the Cross Product Calculator
- Enter the three components of vector a: a1, a2, and a3.
- Enter the three components of vector b: b1, b2, and b3.
- Read the cross product a x b, its magnitude, and the dot product a.b, which update automatically as you type.
- Check the relationship line and the step-by-step panel to see whether the vectors are parallel or perpendicular, and how each value was calculated.
Formula & method
Worked examples
Find the cross product of a=(1, 2, 3) and b=(4, 5, 6).
- i (first) component: a2*b3 - a3*b2 = 2*6 - 3*5 = 12 - 15 = -3
- j (second) component: a3*b1 - a1*b3 = 3*4 - 1*6 = 12 - 6 = 6
- k (third) component: a1*b2 - a2*b1 = 1*5 - 2*4 = 5 - 8 = -3
- Magnitude: |a x b| = sqrt((-3)^2 + 6^2 + (-3)^2) = sqrt(9 + 36 + 9) = sqrt(54) = 7.348469
- Dot product (bonus check): a.b = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32
Result: a x b = (-3, 6, -3), magnitude = 7.348469, dot product = 32. Neither parallel nor perpendicular.
Find the cross product of a=(2, 0, 0) and b=(0, 3, 0), two vectors along the x and y axes.
- i component: a2*b3 - a3*b2 = 0*0 - 0*3 = 0
- j component: a3*b1 - a1*b3 = 0*0 - 2*0 = 0
- k component: a1*b2 - a2*b1 = 2*3 - 0*0 = 6
- Magnitude: |a x b| = sqrt(0^2 + 0^2 + 6^2) = sqrt(36) = 6
- Dot product: a.b = 2*0 + 0*3 + 0*0 = 0, so a and b are perpendicular
Result: a x b = (0, 0, 6), magnitude = 6, dot product = 0. The vectors are perpendicular, and the magnitude 6 equals the area of the 2 by 3 rectangle they form.
Cross product, magnitude, and dot product for common vector pairs
| Vector a | Vector b | a x b | |a x b| | a . b | Relationship |
|---|---|---|---|---|---|
| (1, 2, 3) | (4, 5, 6) | (-3, 6, -3) | 7.348469 | 32 | Neither |
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 | 0 | Perpendicular |
| (0, 1, 0) | (0, 0, 1) | (1, 0, 0) | 1 | 0 | Perpendicular |
| (2, 0, 0) | (0, 3, 0) | (0, 0, 6) | 6 | 0 | Perpendicular |
| (1, 2, 3) | (2, 4, 6) | (0, 0, 0) | 0 | 28 | Parallel |
Cross products of the standard basis vectors i, j, k
| Pair | Cross product | Notes |
|---|---|---|
| i x j | k = (0, 0, 1) | Right-hand rule, forward order |
| j x k | i = (1, 0, 0) | Cycles i, j, k forward |
| k x i | j = (0, 1, 0) | Cycles i, j, k forward |
| j x i | -k = (0, 0, -1) | Reverse order flips the sign |
| k x j | -i = (-1, 0, 0) | Reverse order flips the sign |
| i x k | -j = (0, -1, 0) | Reverse order flips the sign |
Common mistakes to avoid
- Getting the middle (j) component backwards. The pattern for each component skips its own index and subtracts diagonal products, but the middle term is a3*b1 - a1*b3, not a1*b3 - a3*b1. Swapping this term flips its sign and gives a completely wrong resultant vector, even when the first and third components are correct.
- Assuming the cross product is commutative. Unlike ordinary multiplication or the dot product, a x b does not equal b x a. The cross product is anti-commutative: b x a = -(a x b), so reversing the order keeps the same magnitude but reverses every component.
- Confusing the cross product with the dot product. The dot product a.b returns a single number and tests for perpendicularity (a.b = 0 means perpendicular). The cross product a x b returns a vector and tests for parallelism (a x b = zero vector means parallel). Mixing these two checks up leads to the opposite conclusion about the vectors.
- Trying to use the cross product on 2D vectors directly. The cross product formula needs three components per vector. A 2D vector (x, y) has no cross product on its own; treat it as (x, y, 0) first. The result will then always point purely along the z-axis, which is why some textbooks use a shortcut scalar formula, x1*y2 - x2*y1, for 2D problems.
Glossary
- Cross product
- An operation on two 3D vectors, written a x b, that produces a third vector perpendicular to both, with magnitude equal to the area of the parallelogram they form.
- Dot product
- An operation on two vectors that produces a single scalar, a.b = a1b1 + a2b2 + a3b3, used to test for perpendicularity and to measure the angle between vectors.
- Magnitude
- The length of a vector, found as the square root of the sum of the squares of its components. For a cross product, it equals |a| |b| sin(theta).
- Right-hand rule
- A convention for finding the direction of a x b: point the fingers of your right hand along a and curl them toward b, and your thumb points along the cross product.
- Orthogonal (perpendicular) vectors
- Two vectors that meet at a 90 degree angle. Their dot product is always zero, which is the fastest way to test for perpendicularity.
- Parallel vectors
- Two vectors that point in the same or exactly opposite direction, one being a scalar multiple of the other. Their cross product is always the zero vector.
Frequently asked questions
What is the cross product of two vectors?
The cross product of two 3D vectors a and b is a new vector a x b, perpendicular to both a and b, computed as (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1). Its direction follows the right-hand rule, and its magnitude equals the area of the parallelogram the two vectors form.
Is the cross product the same as the dot product?
No. The dot product returns a single number and is zero when two vectors are perpendicular. The cross product returns a whole new vector, perpendicular to the original two, and is the zero vector when the original vectors are parallel. This calculator shows both values together.
What does it mean if the cross product is the zero vector?
A zero cross product means the two vectors are parallel (or one of them is the zero vector), because sin(theta) = 0 only when theta is 0 or 180 degrees. There is no direction perpendicular to two vectors that point along the same line.
How do you find the direction of a x b?
Use the right-hand rule: point the fingers of your right hand in the direction of a, curl them toward b through the smaller angle, and your extended thumb points in the direction of a x b. Swapping the order to b x a reverses that direction exactly.
Can I calculate the cross product of 2D vectors?
The full cross product needs three components, so a 2D vector (x, y) is treated as (x, y, 0). The result always points purely along the z-axis, and many textbooks simplify this special case to a single scalar, x1y2 - x2y1, which represents the signed area between the two vectors.
Is a x b the same as b x a?
No, the cross product is anti-commutative: b x a = -(a x b). Both results have the same magnitude, but every component flips sign because reversing the vector order reverses the direction given by the right-hand rule.