کراس پروڈکٹ کیلکولیٹر – 3D ویکٹر

Calculate the cross product (vector product) of two 3-dimensional vectors. Get the resulting vector, its magnitude, the angle between the original vectors, and a detailed step-by-step solution using the determinant method.

اس کیلکولیٹر کو کیسے استعمال کریں

Vector A = (a₁, a₂, a₃) درج کریں
Vector B = (b₁, b₂, b₃) درج کریں
حساب کریں بٹن پر کلک کریں
کیلکولیٹر کے نیچے دکھائے گئے نتیجے کو پڑھیں

What Is the Cross Product?

The cross product (also called vector product) of two vectors A and B produces a third vector C = A × B that is perpendicular to both A and B. The cross product is only defined in 3-dimensional space (and 7D, as a mathematical curiosity).

The magnitude of the cross product equals the area of the parallelogram formed by the two vectors: |A × B| = |A||B|sin(θ), where θ is the angle between them.

The direction follows the right-hand rule: point fingers in the direction of A, curl them toward B, and the thumb points in the direction of A × B.

Cross Product Formula – Determinant Method

For vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃):

A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

Using the 3×3 determinant with unit vectors î, ĵ, k̂:

|  î   ĵ   k̂  |
| a₁  a₂  a₃ |
| b₁  b₂  b₃ |

= î(a₂b₃ - a₃b₂) - ĵ(a₁b₃ - a₃b₁) + k̂(a₁b₂ - a₂b₁)

Example: A = (1, 2, 3), B = (4, 5, 6)
A × B = (2×6−3×5, 3×4−1×6, 1×5−2×4) = (12−15, 12−6, 5−8) = (−3, 6, −3)

Properties of the Cross Product

Anti-commutativity: A × B = −(B × A). The cross product reverses direction when operands are swapped.
Zero for parallel vectors: If A and B are parallel (or anti-parallel), A × B = 0 (zero vector). sin(0°) = sin(180°) = 0.
Maximum for perpendicular vectors: |A × B| is maximum when A ⊥ B (θ = 90°, sin(90°) = 1).
Distributive: A × (B + C) = (A × B) + (A × C)
Scalar multiplication: (kA) × B = k(A × B)
NOT associative: A × (B × C) ≠ (A × B) × C in general

Cross Product vs Dot Product

Property	Cross Product (A × B)	Dot Product (A · B)
Result type	Vector (3D)	Scalar (number)
Formula	(a₂b₃−a₃b₂, ...)	a₁b₁ + a₂b₂ + a₃b₃
Geometric meaning	Area of parallelogram, perpendicular vector	Projection, work done
Zero when	Vectors are parallel	Vectors are perpendicular
Commutativity	Anti-commutative: A×B = −B×A	Commutative: A·B = B·A

Applications of the Cross Product

Torque: τ = r × F (torque = position vector × force). Fundamental in rotational mechanics.
Angular momentum: L = r × p (position × momentum)
Surface normals: In 3D graphics, the cross product of two edge vectors gives the normal to a polygon face — used for lighting and shading calculations
Magnetic force: F = qv × B (force on a charge in a magnetic field)
Area of triangle/parallelogram: Area of parallelogram = |A × B|; area of triangle = |A × B|/2
Testing collinearity: Three points are collinear if the cross product of the vectors between them is zero

Unit Vectors and the Cross Product

The standard basis vectors have these cross product relationships:

î × ĵ = k̂
ĵ × k̂ = î
k̂ × î = ĵ
î × î = ĵ × ĵ = k̂ × k̂ = 0 (zero vector)

These relationships follow cyclically: i→j→k→i in the positive direction. Reversing gives the negative: ĵ × î = −k̂.

Finding the Angle Between Two Vectors

You can find the angle θ between two vectors using either the dot or cross product:

Using dot product: cos θ = (A · B) / (|A||B|)
Using cross product: sin θ = |A × B| / (|A||B|)

The dot product method is more commonly used because it directly gives θ = arccos(A·B / |A||B|) for any angle 0°–180°. The cross product gives the same angle but you need arcsin, which only covers 0°–90° (ambiguous for obtuse angles).

آخری اپ ڈیٹ: March 2026

Frequently Asked Questions

What is the cross product of two parallel vectors?

Zero (the zero vector). When two vectors are parallel, the angle between them is 0° or 180°, and sin(0) = sin(180) = 0, so |A × B| = |A||B|sin(θ) = 0.

Is the cross product commutative?

No. The cross product is anti-commutative: A × B = −(B × A). Swapping the order reverses the direction of the resulting vector. This is why order matters in cross product calculations.

What is the cross product used for in physics?

Torque (r × F), angular momentum (r × p), magnetic force on a moving charge (qv × B), and angular velocity calculations all use the cross product. Any physical quantity involving rotation or a right-angle relationship between vectors likely involves the cross product.

Can you take the cross product of 2D vectors?

Not directly, but you can extend 2D vectors to 3D by adding z = 0: A = (a₁, a₂, 0) and B = (b₁, b₂, 0). The cross product is then (0, 0, a₁b₂ − a₂b₁) — a vector pointing in the z direction. The z component (a₁b₂ − a₂b₁) is the "2D cross product" scalar.

How do you verify a cross product answer?

Check that A × B is perpendicular to both A and B by computing the dot products: (A × B) · A should equal 0, and (A × B) · B should equal 0. If both are zero (or very small due to floating-point), your cross product is correct.

What is the magnitude of the cross product?

|A × B| = |A||B|sin(θ), where θ is the angle between A and B. This equals the area of the parallelogram formed by the two vectors. The triangle with A and B as sides has area = |A × B|/2.

What does A × A equal?

A × A = 0 (zero vector). Any vector crossed with itself gives the zero vector because the angle θ = 0°, and sin(0°) = 0. Equivalently, the parallelogram collapses to a line with zero area.

How is the cross product related to the right-hand rule?

Point your right hand's fingers in the direction of A, then curl them toward B. Your thumb points in the direction of A × B. This mnemonic establishes the orientation convention — the cross product follows a right-handed coordinate system.

What is the triple product?

The scalar triple product A · (B × C) gives the volume of the parallelepiped formed by three vectors. If this equals zero, the three vectors are coplanar. The vector triple product A × (B × C) = B(A·C) − C(A·B) (the BAC-CAB rule).

Why doesn't the cross product work in 2D?

The cross product requires a third dimension for the result vector to be perpendicular to both inputs. In 2D, there's no third direction available. Mathematically, the cross product is only defined (as a vector) in 3D and 7D.