Enter two 3D vectors to find the perpendicular vector, vector magnitude, angle between vectors, and the area tied to your result in one place.
Enter vector components and click Calculate to see results
Type the x, y, and z components for your first vector exactly as they appear in the problem.
Add the second 3D vector using whole numbers, decimals, or negative values.
Click the button to find the cross product with the determinant method applied automatically.
Review the perpendicular vector, vector magnitude, angle, and relationship between the inputs.
You can use this calculator any time you need the cross product of two vectors in three-dimensional space. Start by entering the x, y, and z components for Vector A and Vector B. The tool accepts whole numbers, decimals, and negative values, so you can work through classroom problems, engineering homework, or quick checks during design work.
After you click Calculate Cross Product, the result panel shows the new perpendicular vector, the vector magnitude of each input, the magnitude of the cross product, and the angle between vectors. That gives you the full picture at once instead of making you run separate calculations for geometry, direction, and area.
If you are double-checking a manual solution, use the calculator after you work the problem by hand. This is a smart way to catch a sign error in the determinant method, which is one of the most common cross product mistakes. If the result is the negative of what you expected, the issue is usually vector order rather than arithmetic.
Use it when you want a fast answer for a homework set, a physics lab, a CAD model, or a game development task that needs a unit normal vector or surface direction. It also works well when you want to compare your hand work against the live result and build confidence with coordinate geometry, signs, orientation, and the right-hand rule.
The main answer is the cross product vector A x B. This vector is perpendicular to both original vectors, which means it points at a 90 degree angle to the plane that contains them. In graphics, that direction can act as a surface normal. In physics, it can describe a moment direction such as torque or angular momentum.
The magnitude of the cross product is just as important as its direction. It equals |A||B|sin(theta), so it measures the parallelogram area spanned by the two vectors. If you are working with a triangle instead of a parallelogram, divide the magnitude by 2. That one relationship makes the cross product useful for geometry, mechanics, and mesh calculations.
The angle shown in the results helps you judge how close your vectors are to perpendicular or parallel. A larger sine value means a larger perpendicular component. When the angle is near 90 degrees, the cross product magnitude is close to its maximum for those vector lengths. When the angle is near 0 degrees or 180 degrees, the cross product shrinks toward zero.
If you see a zero vector, that usually means the vectors are parallel, anti-parallel, or one vector is the zero vector. The calculator also flags whether the vectors are parallel or perpendicular, so you can interpret the result without doing another check.
For vectors A = (ax, ay, az) and B = (bx, by, bz), the determinant method gives you:
A x B = (aybz - azby, azbx - axbz, axby - aybx)
You can think of this as expanding a 3 by 3 determinant with the unit vectors i, j, and k in the first row. The middle component gets a sign change during expansion, which is why many students make an error there. A calculator helps you verify the arithmetic, but it also helps to remember that the cross product is anti commutative. If you swap the vector order, you flip the sign of the whole answer.
Suppose you want the cross product of A = (2, -1, 3) and B = (4, 5, -2). Plug the values into the formula one component at a time:
So the cross product is (-13, 16, 14). The magnitude is sqrt(621) ≈ 24.92. That means the parallelogram area formed by A and B is about 24.92 square units. If you were finding the area of a triangle from the same sides, the answer would be about 12.46 square units.
This same setup lets you estimate the angle between vectors because |A x B| = |A||B|sin(theta). Once you know the vector magnitude of each input, you can solve for theta and check whether the vectors are close to perpendicular.
If a wrench applies a force of F = (0, 40, 0) N at a position vector of r = (0.3, 0, 0) m, then torque is r x F = (0, 0, 12). The magnitude is 12 N·m. This tells you both how strong the turning effect is and which axis it points around.
Two edge vectors on a surface might be u = (3, 0, 0) and v = (0, 2, 0). Their cross product is (0, 0, 6). When you convert that to a unit normal vector, you get a clean direction for shading, face culling, and lighting in a 3D scene.
Suppose two frame edges are (6, 0, 0) and (0, 4, 0). The cross product is (0, 0, 24), so the parallelogram area is 24 square units. If those vectors describe a triangular brace, the area is 12 square units.
With A = (1, 2, 0) and B = (2, 1, 4), the cross product is (8, -4, -3). The sign pattern helps you confirm orientation and handedness when you build a local coordinate system from sensor or motion data.
In electromagnetism, force can depend on v x B. If a particle moves with v = (5, 0, 0) through a magnetic field B = (0, 0, 0.2), then v x B = (0, -1, 0). Multiply by charge to get the force vector direction and size.
Tip: if your answer has the right magnitude but the wrong direction, check the vector order first. That fixes many homework and coding mistakes faster than redoing the whole determinant.
One major idea many calculator pages skip is the geometry behind the answer. The cross product is not just another vector operation. It connects direction, area, and orientation in a way that makes it very useful across math and science. When you see the result vector, you are really seeing the normal direction to the plane formed by your two inputs.
The magnitude of that normal vector equals the parallelogram area. If the vectors spread wide apart, the sine term grows and the area gets larger. If they lie almost on top of each other, the sine term gets small and the area collapses. This is why the cross product is a clean way to measure how much one vector sticks out sideways from another.
Direction comes from the right hand rule. Point your fingers along the first vector, curl toward the second vector, and your thumb shows the result. That rule matters in physics and computer graphics because the sign of the normal changes visible behavior. A flipped normal can turn a front-facing surface into a back-facing one, or reverse the expected direction of torque.
In 2D, you do not usually talk about a full cross product vector. Instead, you often use the scalar value axby - aybx. That number acts like a signed area and helps with orientation tests, polygon winding, and line segment problems. So while the standard cross product belongs to 3D space, the same idea still appears in 2D coordinate geometry in a slightly different form.
If you move on to the scalar triple product, you combine a dot product with a cross product to measure volume. That makes this calculator a strong foundation tool for later topics in linear algebra, vector calculus, and 3D modeling.
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A cross product gives you a new vector that is perpendicular to both input vectors. Its magnitude equals the area of the parallelogram formed by the two vectors, and its direction follows the right-hand rule.
Write the vectors in component form, place unit vectors i, j, and k across the top of a determinant, and expand the determinant. For vectors A = (ax, ay, az) and B = (bx, by, bz), the result is (aybz - azby, azbx - axbz, axby - aybx).
The standard vector cross product is defined for three-dimensional space because it returns a vector that is perpendicular to both inputs. In 2D, you usually work with a scalar value that represents signed area instead of a full perpendicular vector.
A zero cross product means the vectors are parallel, anti-parallel, or one vector has zero length. In each case, the angle between them produces no perpendicular area.
Take the magnitude of the cross product. That value is the area of the parallelogram formed by the two vectors. If you need the area of a triangle, divide that magnitude by 2.
A cross product returns a vector and measures perpendicular area, while a dot product returns a scalar and measures alignment or projection. The cross product uses sine of the angle, and the dot product uses cosine.
Yes. The cross product is anti-commutative, which means A x B = -(B x A). Reversing the order flips the direction of the result but keeps the same magnitude.
Point your right-hand fingers in the direction of the first vector and curl them toward the second vector. Your thumb points in the direction of the cross product vector.