Definition of the three-dimensional cross product

For ordered vectors A = ⟨A_x, A_y, A_z⟩ and B = ⟨B_x, B_y, B_z⟩, the cross product A × B is a vector perpendicular to the plane spanned by A and B. Its direction follows the right-hand rule, and its magnitude is the area of their parallelogram.

Default component calculation

For A = ⟨1, 2, 3⟩ and B = ⟨4, 5, 6⟩, the x component is 2×6 − 3×5 = −3. The y component is 3×4 − 1×6 = 6. The z component is 1×5 − 2×4 = −3. Therefore A × B = ⟨−3, 6, −3⟩.

The determinant-style panel preserves the component order. A common hand-calculation error is to forget the middle cofactor’s sign. Writing the y expression directly as A_zB_x − A_xB_z avoids applying a second accidental negative.

Direction and the right-hand rule

Point the fingers of the right hand along A and curl them toward B through the smaller angle. The thumb points in the A × B direction. This identifies one of the two normals to the plane. OpenStax and NASA/JPL materials use the right-hand convention for vector orientation and rotations.

The rule assumes A and B share a tail and the coordinate system is right-handed. On a two-dimensional screen, y may increase downward, and some graphics systems use different handedness or vertex winding. Translate the coordinate convention before applying a visual mnemonic.

Order matters: anti-commutativity

Swapping the inputs leaves magnitude unchanged but reverses every component. In the default, B × A = ⟨3, −6, 3⟩. The swap button performs that input change so the direction reversal is visible in the main result rather than merely stated.

This property is why the cross product is not ordinary multiplication. A × B and B × A represent opposite orientations. In torque, surface normals, winding, and magnetic-force formulas, reversing order can reverse the physical or geometric direction.

Magnitude, angle, and area

The default magnitude is √[(−3)² + 6² + (−3)²] = √54, approximately 7.348469. It equals the area of the parallelogram formed by A and B. A triangle using the two vectors as sides has half that area, approximately 3.674235.

The angle is calculated from the dot product with a clamped cosine ratio to protect against tiny floating-point excursions outside −1 to 1. For the default, A · B = 32 and the smaller angle is about 12.933154 degrees. The cross magnitude is small relative to |A||B| because the vectors point in similar directions.

Unit normal

Dividing A × B by its magnitude produces a unit normal. The default unit vector is approximately ⟨−0.408248, 0.816497, −0.408248⟩. It has length one and retains the A-to-B right-hand orientation.

A unit normal cannot be formed when the cross product is zero. Parallel vectors, antiparallel vectors, and any pair containing a zero vector do not define a unique oriented plane normal. The calculator reports the degeneracy instead of dividing by zero.

Orthogonality checks

A correct cross product must have dot product zero with both inputs. In the default, ⟨−3,6,−3⟩ · ⟨1,2,3⟩ = −3 + 12 − 9 = 0. With B, the result is −12 + 30 − 18 = 0.

For decimal inputs, the checks may be a tiny nonzero value because of floating-point arithmetic. The entered tolerance classifies them for the status sentence, but the raw calculated values remain displayed. Choose tolerance relative to the problem’s scale.

Parallel and nearly parallel inputs

If B is a scalar multiple of A, their angle is zero or 180 degrees, sine is zero, and the cross product is the zero vector. Geometrically, the would-be parallelogram collapses to a line. A near-zero cross magnitude can indicate near parallelism.

Absolute tolerance alone can be misleading for huge or tiny coordinates. A scale-aware check compares |A × B| with |A||B|. Production geometry code should use relative tolerances, input normalization when appropriate, and robust predicates when topology depends on the sign.

Units of the result

Cross-product magnitude multiplies the input units. Two displacement vectors measured in meters produce an area magnitude in square meters. Position in meters crossed with force in newtons produces torque units of newton-meters. Direction remains a vector orientation.

A newton-meter of torque shares dimensions with a joule but represents a different physical quantity. Do not relabel cross-product output based only on dimensional similarity. Identify the formula and context.

Applications

Cross product versus dot product

The dot product returns a scalar and measures aligned projection through cosine. The cross product returns a 3D vector and measures perpendicular oriented area through sine. Both appear in the result because together they help recover angle and check geometry.

This calculator is deliberately three-dimensional. Generalized exterior products and seven-dimensional cross products are different mathematical constructions. Do not append zeros to arbitrary high-dimensional data and assume the 3D result preserves the intended geometry.

Result auditing workflow

First verify the coordinate order and units. Next recompute the three components, check B × A is the negative, and dot the result with both inputs. Confirm the magnitude equals the square root of component squares. Finally, interpret the sign under the declared handedness.

A shared numeric interface does not imply shared formulas.

Scaling laws and sign tests

Multiplying A by a scalar c multiplies A × B by c. A negative scalar also reverses direction. Scaling both vectors by c and d multiplies the cross product by cd, so area changes by the absolute value of cd while orientation flips only when the product cd is negative. These properties provide quick error checks before repeating all three component calculations.

For example, doubling A in the default should change ⟨−3,6,−3⟩ to ⟨−6,12,−6⟩ and double the parallelogram area. Replacing B with −B should produce ⟨3,−6,3⟩. Adding a multiple of A to B leaves A × B unchanged because A × (B + cA) = A × B + c(A × A), and A × A is zero. That last identity explains why sliding one side of a parallelogram parallel to the other preserves its base-height area.

Worked torque interpretation

Suppose a position vector from a pivot is r = ⟨0.30,0,0⟩ meters and the applied force is F = ⟨0,40,0⟩ newtons. Then r × F = ⟨0,0,12⟩ newton-meters. The positive z direction follows the right-hand rule: fingers point along the positive x lever arm and curl toward the positive y force. The magnitude also equals rF sin 90°, or 0.30 × 40 = 12.

If the force is reversed, the torque becomes ⟨0,0,−12⟩. If the force points along the lever arm, the sine of the angle is zero and so is torque, even though both vectors are nonzero. A mechanical analysis must still account for the actual pivot, coordinate frame, force application point, and sign convention. The calculator supplies vector algebra, not a structural or safety evaluation.

Choosing a tolerance responsibly

The default tolerance is one-billionth in the numeric units of the computed checks. It is convenient for the modest integer example, but no absolute threshold fits every dataset. Coordinates from surveying, graphics meshes, sensors, or orbital calculations can differ by many orders of magnitude.

A defensible workflow relates tolerance to |A||B|, measurement uncertainty, and downstream consequences. Record whether inputs were normalized and whether the classification is meant to detect exact symbolic parallelism or practical near-parallelism. Do not increase tolerance merely to force a desired “perpendicular” or “parallel” label; investigate scale and data quality first.

Frequently asked questions

References

OpenStax University Physics — Products of vectors and right-hand rule

OpenStax Calculus Volume 3 — The cross product

NASA/JPL Navigation and Ancillary Information Facility — Rotation and right-hand conventions