Introduction
The cross product calculator helps you find the cross product of two vectors in seconds. The cross product is a math operation that takes two vectors and gives you a new vector that points straight up from both of them. It is one of the most useful tools in linear algebra, physics, and engineering.
With this calculator, you can enter 3D or 2D vectors using numbers, fractions, or even symbols like a, b, and c. It gives you the full result in component form and î, ĵ, k̂ notation. You also get the magnitude, the angle between the vectors, the parallelogram area, the triangle area, the dot product, and the unit normal vector. A clear, step-by-step solution shows you exactly how the answer is found using the determinant method.
The tool also draws a 3D vector diagram, a right-hand rule chart, and a bar graph that compares each component side by side. You can check your own work with the built-in answer verifier. Whether you are a student learning vectors for the first time or solving a homework problem, this cross product calculator makes the work fast and simple.
How to Use Our Cross Product Calculator
Enter two vectors and this calculator will find their cross product. You will get the result in component form and i, j, k notation, a step-by-step solution, the magnitude, the angle between the vectors, and helpful diagrams.
Input Method: Pick how you want to type your vectors. Choose Structured Fields to enter each part of the vector in its own box. Choose Expression to type the full problem in one line, like (1, 2, 3) × (4, 5, 6).
Dimension: Pick 3D if your vectors have three parts (x, y, z). Pick 2D if they only have two parts (x, y). In 2D mode, the result is a single number along the z-axis.
Vector A: Enter the x, y, and z values for your first vector. You can use whole numbers, decimals, fractions like 3/4, or even letters like a for symbolic math.
Vector B: Enter the x, y, and z values for your second vector using the same format as Vector A.
Quick Presets and Examples: Click any preset or example button to load a ready-made problem and see how the calculator works.
Prefer Exact Fractions: Turn this on to show results as fractions instead of decimals. This is useful when your inputs are fractions.
Round Decimals: Turn this on and choose how many decimal places you want in the output.
Show Angle θ: Check this box to see the angle between Vector A and Vector B in degrees and radians.
Show Right-Hand-Rule Diagram: Check this box to display a picture that shows which way the cross product points using the right-hand rule.
Show 3D Vector Diagram: Check this box to see all three vectors drawn on a 3D set of axes. This only works in 3D mode.
Calculate: Press the Calculate button to get your answer. Press Reset to clear everything and start over.
Check Your Answer: Type your own answer into the verify box and click Verify to see if it matches the correct cross product.
What Is a Cross Product?
The cross product is a math operation that takes two vectors and gives you a brand new vector. This new vector points in a direction that is perpendicular (at a right angle) to both of the original vectors. Think of it like this: if two arrows lie flat on a table, their cross product points straight up from the table.
How the Cross Product Works
To find the cross product of two 3D vectors A = ⟨a₁, a₂, a₃⟩ and B = ⟨b₁, b₂, b₃⟩, you use this formula:
- x-component: a₂ · b₃ − a₃ · b₂
- y-component: a₃ · b₁ − a₁ · b₃
- z-component: a₁ · b₂ − a₂ · b₁
This comes from finding the determinant of a 3×3 matrix made up of the unit vectors î, ĵ, k̂ and the components of your two vectors.
Key Properties of the Cross Product
The cross product has a few important rules:
- Anti-commutative: Swapping the order flips the sign. A × B = −(B × A).
- Perpendicular result: The answer is always at a right angle to both input vectors.
- Parallel vectors give zero: If two vectors point in the same or opposite direction, their cross product is the zero vector ⟨0, 0, 0⟩.
- Direction follows the right-hand rule: Curl the fingers of your right hand from A toward B. Your thumb points in the direction of A × B.
What the Magnitude Tells You
The length (magnitude) of the cross product equals the area of the parallelogram formed by the two vectors. Half of that value gives you the area of the triangle they form. The formula is |A × B| = |A| · |B| · sin(θ), where θ is the angle between A and B. You can use our trig calculator to evaluate the sine function for specific angles.
2D Cross Product
In two dimensions, vectors have no z-component. The cross product simplifies to a single number: a₁ · b₂ − a₂ · b₁. This scalar tells you the signed area of the parallelogram. A positive value means B is counterclockwise from A, and a negative value means B is clockwise from A.
Where Cross Products Are Used
Cross products show up in physics and engineering all the time. They help calculate torque (rotational force), the magnetic force on a moving charge, and surface normals in 3D graphics. Any time you need a direction that is perpendicular to a surface or a plane, the cross product is the tool to use. The closely related dot product is another essential vector operation that measures how much two vectors point in the same direction rather than producing a perpendicular vector.