Kalkulator Hasil Darab Silang – Vektor 3D
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.
Cara menggunakan kalkulator ini
- Masukkan Vector A = (a₁, a₂, a₃)
- Masukkan Vector B = (b₁, b₂, b₃)
- Klik butang Kira
- Baca keputusan yang dipaparkan di bawah kalkulator
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).
Kemas kini terakhir: 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.