Calculator Inputs
Formula Used
For vectors A = (a1, a2, a3) and B = (b1, b2, b3), the cross product is:
A × B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
The magnitude is |A × B|. It equals the area of the parallelogram formed by both vectors.
The triangle area is |A × B| ÷ 2. The unit normal is found by dividing the cross product by its magnitude.
The angle is calculated with atan2(|A × B|, A · B). This helps handle acute, right, and obtuse angles.
How to Use This Calculator
- Enter the x, y, and z components for vector A.
- Enter the x, y, and z components for vector B.
- Choose A × B or B × A.
- Add an optional scale value if needed.
- Select decimal places for the answer.
- Press the calculate button.
- Review the vector, magnitude, area, angle, and unit normal.
- Download the result as CSV or PDF.
Example Data Table
| Vector A | Vector B | A × B | Magnitude | Use Case |
|---|---|---|---|---|
| (1, 2, 3) | (4, 5, 6) | (-3, 6, -3) | 7.3485 | Component practice |
| (3, 0, 0) | (0, 4, 0) | (0, 0, 12) | 12.0000 | Area check |
| (2, -1, 4) | (1, 3, -2) | (-10, 8, 7) | 14.5945 | Engineering vector |
| (0, 1, 0) | (0, 0, 1) | (1, 0, 0) | 1.0000 | Right-hand rule |
About the Cross Product Calculator
Purpose
A cross product creates a new vector from two three-dimensional vectors. The new vector is perpendicular to both inputs. This calculator helps students, teachers, designers, and engineers check vector work quickly. It accepts positive values, negative values, decimals, and zero components. It also shows the steps, so the answer is easier to verify.
Why It Matters
The cross product is useful when direction matters. It appears in physics, geometry, mechanics, graphics, and many engineering tasks. Torque, angular momentum, surface normals, and oriented area often use this operation. A simple component answer is helpful, but a full result gives more context. This tool returns the vector, magnitude, unit normal, angle, triangle area, and parallelogram area.
Direction and Area
Direction is controlled by order. A × B points one way. B × A points the opposite way. The calculator includes both choices because sign errors are common in vector problems. The right-hand rule can be used to check the direction. Curl your fingers from the first vector toward the second vector. Your thumb shows the positive normal direction.
Magnitude and Angle
The magnitude of the cross product equals |A||B|sinθ. It also equals the area of the parallelogram made by both vectors. Half of that value gives the triangle area. The angle result is calculated from the cross product magnitude and the dot product. This gives a stable angle check for many vector pairs.
Better Checking
The detailed output helps catch mistakes. You can compare each component with the displayed expansion. You can also review the dot product, the unit normal, and the parallel warning. If the cross product is zero, the vectors may be parallel, anti-parallel, or one input may be the zero vector. Use the CSV or PDF export when you need to save results for homework, reports, lab notes, or project records.
FAQs
What is a cross product?
A cross product is a vector operation for two 3D vectors. It creates a new vector perpendicular to both input vectors.
Does vector order matter?
Yes. A × B and B × A have opposite directions. Their magnitudes are the same, but their signs are reversed.
Can this calculator handle decimals?
Yes. You can enter whole numbers, decimals, negative values, and zero values for each vector component.
What does the magnitude mean?
The magnitude is the length of the cross product vector. It also equals the parallelogram area formed by the two vectors.
How is triangle area calculated?
The triangle area is one half of the cross product magnitude. This works when the two vectors form adjacent triangle sides.
What is a unit normal?
A unit normal is the cross product direction with length one. It is found by dividing the vector by its magnitude.
What if the cross product is zero?
A zero cross product means the vectors are parallel, anti-parallel, or one vector has zero length.
Why include CSV and PDF downloads?
Downloads help you save calculations for assignments, design notes, reports, or later checking without retyping the values.