Cross Product Calculator – Complete Guide to A × B, Normal Vector, Area and Triple Products
The Cross Product Calculator on MyTimeCalculator is built for 3D vector problems in mathematics, physics, engineering, computer graphics and geometry. It computes A × B, magnitudes, unit normal vectors, angles, areas, scalar triple products, vector triple products and volumes in a single consistent interface.
In three dimensions, the cross product captures the idea of a vector perpendicular to a plane, with magnitude equal to an area. Combined with the dot product, it gives a powerful toolkit for understanding both direction and size of geometric and physical quantities.
1. Cross Product A × B
For vectors A = <Ax, Ay, Az> and B = <Bx, By, Bz>, the cross product is defined as:
The Cross Product Calculator computes these components, formats the result as <x, y, z> and also provides the magnitude |A × B|. This vector is perpendicular to both A and B and follows the right-hand rule.
2. Magnitudes and Unit Normal Vector
The magnitudes of A and B are computed with the Euclidean norm:
Once A × B is known, its magnitude |A × B| is:
The unit normal vector n̂ is obtained by dividing the cross product by its magnitude (when non-zero):
This normal vector is widely used for defining orientations of planes, surface normals in 3D graphics and perpendicular directions in mechanics and engineering designs.
3. Angle Between Vectors and Area
The dot and cross products together connect the angle θ between A and B with their magnitudes:
The calculator uses the cross product to find:
- θ in degrees, using the relationship between dot, magnitudes and arccos.
- The area of the parallelogram spanned by A and B, equal to |A × B|.
- The area of the triangle spanned by A and B, which is ½|A × B|.
These geometric quantities are central in many problems: surface area, torque models, geometric projections and more.
4. Scalar Triple Product and Volume
Given three vectors A, B and C, the scalar triple product is defined as:
It can also be expressed as the determinant of the 3×3 matrix with rows (or columns) A, B and C. Geometrically:
- [A B C] is the signed volume of the parallelepiped determined by A, B and C.
- |[A B C]| is the actual volume.
- If [A B C] = 0, the vectors are coplanar.
The Triple Products tab computes [A B C] and |[A B C]| directly, giving an immediate view of both orientation and volume associated with the three vectors.
5. Vector Triple Products
Vector triple products involve a cross product nested inside another cross product, such as:
They appear in vector identities and advanced mechanics. While there are algebraic identities that simplify them, the calculator evaluates them directly using cross product formulas and returns:
- A × (B × C)
- (A × B) × C
Both results are displayed as 3D vectors, making it easier to compare them and explore vector identities numerically.
6. Determinant Matrix Representation
The cross product is often written as a determinant involving unit vectors i, j, k:
The Determinant Matrix tab displays a matrix with i, j, k in the header row and the components of A and B in the rows beneath. It also summarizes the resulting cross product, helping you connect the symbolic determinant form with the numeric output.
How to Use the Cross Product Calculator
- Choose the appropriate tab: core cross product, triple products, angle and area, or determinant view.
- Enter the components of A, B and C (where required) in the input fields provided.
- Click the “Compute” button for that tab.
- Read the vector and scalar results in the result cards or matrix table.
If any input is missing or invalid, the calculator shows an alert so you can correct the values. When computations are not defined (for example, normal vector for a zero cross product), a clear message appears.
Applications of Cross Product and Triple Products
- Physics: torque τ = r × F, angular momentum, rotational effects, electromagnetic forces.
- Engineering: moments about a point, structural analysis, beam and frame calculations.
- Computer graphics: surface normals, lighting calculations, back-face culling, orientation.
- Robotics: orientation of links, end-effector normals, path planning in 3D space.
- Geometry: areas and volumes of parallelograms, triangles and parallelepipeds.
- Education: visualizing vector operations, checking textbook exercises, exploring vector identities.
Related Tools from MyTimeCalculator
Cross Product Calculator FAQs
Frequently Asked Questions
Explore common questions about cross product, normal vectors, areas, scalar triple products and volumes.
The cross product A × B is a vector perpendicular to both A and B. Its direction follows the right-hand rule and its magnitude equals the area of the parallelogram formed by A and B. This connects orientation and area into a single geometric quantity used in many 3D applications.
The dot product produces a scalar and measures directional alignment, while the cross product produces a vector perpendicular to both inputs and measures rotational or perpendicular aspects. In this calculator, the cross product is used for normals and areas, while the dot product (used internally in some formulas) relates to angles and projections.
A zero cross product means that the vectors are parallel or one of them is the zero vector. In that case, there is no unique perpendicular direction and the area spanned by the two vectors is zero. The calculator recognizes this and indicates that the unit normal vector is undefined in this situation.
The volume comes from the scalar triple product [A B C] = A · (B × C). The signed value gives the oriented volume, and its absolute value gives the actual volume. A zero scalar triple product indicates that the vectors are coplanar and the volume is zero. The Triple Products tab computes both the signed value and its magnitude for you.
Vector triple products such as A × (B × C) and (A × B) × C are not generally equal, and they appear in many vector identities and mechanical formulas. The calculator displays both to help you explore these identities numerically and see how the two expressions differ for given vectors A, B and C.
Writing the cross product as a determinant with the first row being the unit vectors i, j, k and the next two rows being the components of A and B is a compact symbolic way to remember and derive the component formulas. The Determinant tab visualizes this matrix and connects it to the numeric cross product vector computed by the calculator.
Yes. Torque is defined as τ = r × F, where r is the position vector and F is the force. By entering r as A and F as B, you can compute τ, its magnitude and the associated normal direction. The same approach works for other moment and rotational calculations in mechanics and engineering design problems.
Yes. You can freely enter negative values and decimals for any component. The calculator follows the standard cross product, dot product and norm formulas and formats results with a sensible number of decimal places for clarity. Invalid or missing entries trigger an alert so you can correct them quickly.
Yes. The classic cross product with a perpendicular vector result is defined only in three dimensions (and in a special form in seven dimensions in advanced mathematics). This calculator focuses on the standard 3D version used in physics, engineering and geometry courses.