What is the cross product?

The cross product (also called the vector product) is a way to multiply two vectors in 3D and get a new vector that is perpendicular to both. If you’ve seen the dot product before, here’s the key difference: the dot product produces a scalar (a single number), while the cross product produces a vector (a direction + magnitude).

The cross product is written as a × b. If a = (ax, ay, az) and b = (bx, by, bz), then the cross product is:

That is, the components are:

• cx = ay·bz − az·by
• cy = az·bx − ax·bz
• cz = ax·by − ay·bx

The magnitude of the cross product, |a × b|, has a powerful geometric meaning: it equals the area of the parallelogram formed by the two vectors. If the angle between the vectors is θ (theta), then:

This is why the cross product shows up everywhere: in physics it’s behind torque (lever arm × force), in engineering it helps you find a surface normal, and in 3D graphics it’s a standard way to compute the direction a polygon should face.

• Step 1: Read your inputs for a and b.
• Step 2: Compute (cx, cy, cz) using the component formulas above.
• Step 3: Compute magnitude |c| = √(cx² + cy² + cz²).
• Step 4: If |c| ≠ 0, compute the unit normal n = c/|c|.
• Step 5: (Optional) Compute the angle θ between a and b using the dot product identity: cos(θ) = (a·b)/(|a||b|) when both vectors have nonzero length.

The cross product is anti‑commutative, which is a fancy way of saying: a × b = −(b × a). So if you swap the vectors, the magnitude stays the same, but the direction flips. In practical terms: if your normal points “the wrong way,” try switching the order.

How to interpret your result

Once you compute c = a × b, you’ll see a vector like (cx, cy, cz). Here’s what it is telling you:

The cross product direction is perpendicular to the plane that contains a and b. If you draw both vectors from the same starting point, they form a flat plane. The cross product points straight “out” of that plane. The right‑hand rule picks which side is positive.

If a and b are perfectly parallel, the angle θ is 0° or 180°, so sin(θ) = 0 and the magnitude becomes 0. That means there is no unique perpendicular direction: the plane collapses into a single line. As the vectors become more perpendicular, sin(θ) increases and so does the magnitude. The maximum happens at 90°, where sin(90°) = 1, so |a × b| = |a||b|.

If you place vectors a and b tail‑to‑tail, they form a parallelogram. The area of that parallelogram is |a × b|. If you want the area of the triangle formed by a and b instead, it’s simply half: Area_triangle = |a × b| / 2.

In 2D, you can still use the cross product by embedding the vectors into 3D: set z = 0 for both. Then the cross product will point purely in the z direction (positive or negative). That sign tells you whether the rotation from a to b is counter‑clockwise (+z) or clockwise (−z), using the right‑hand rule.

Worked examples (with explanations)

Examples are where the cross product finally feels “real,” because you can see the geometry. Here are a few common patterns you’ll run into in school and real applications.

Let a = (3, 0, 0) and b = (0, 4, 0). These are perpendicular. Compute:

Interpretation: the result points in +z and the magnitude is 12. The parallelogram area is 12, and the triangle area would be 6. This matches what you’d expect: base 3 times height 4 equals 12.

Let a = (2, 2, 2) and b = (4, 4, 4). Vector b is just 2 times a, so they point in the same direction (parallel). The cross product becomes the zero vector: a × b = (0, 0, 0).

Interpretation: there’s no “area” between them because they don’t form a real parallelogram—just a line. In physics terms, if a is a lever arm and b is a force, a force perfectly along the lever produces no torque.

Let a = (1, 2, 3) and b = (4, 5, 6). Then:

Interpretation: (−3, 6, −3) is perpendicular to both a and b. The magnitude ~7.35 is the parallelogram area. If you normalize it, you get a unit normal direction that is often used in geometry and graphics.

In physics, torque is τ = r × F, where r is the lever arm vector (from pivot to where force is applied) and F is the force vector. The magnitude is |τ| = |r||F|sin(θ), so only the part of the force that is perpendicular to r creates torque. If the force is applied straight along the lever (θ = 0), torque is 0.

Frequently Asked Questions

• Can I use this for 2D vectors?

  Yes. Set z = 0 for both vectors. The cross product will point in the ±z direction. The magnitude will still represent the parallelogram area between the 2D vectors.

• Why is the cross product magnitude the area?

  The area of a parallelogram is base times height. If you treat |a| as the base, the height is the component of b perpendicular to a, which is |b|sin(θ). Multiply them: |a||b|sin(θ), which is exactly |a × b|.

• What if one vector is the zero vector?

  If a = (0,0,0) or b = (0,0,0), then the cross product is (0,0,0). There’s no direction, and the area is 0. The angle between vectors is also undefined because you can’t divide by a zero length.

• Why does changing the order flip the sign?

  The cross product follows the right‑hand rule. Swapping the order changes the rotation direction, so the perpendicular direction reverses. That’s why a × b = −(b × a).

• How do I know if my answer is correct?

  Quick check: the cross product should be perpendicular to both vectors. If you take the dot product c·a and c·b, both should be 0 (up to rounding). This calculator also shows the direction and magnitude to help you sanity‑check.

• Where is the cross product used in real life?

  Physics (torque, angular momentum), engineering (surface normals), robotics (orientation), and 3D graphics (lighting and face normals). Anywhere you need a perpendicular direction or an area spanned by two vectors, the cross product shows up.