Vector Calculator 2D & 3D (Dot, Cross & Triple Product)
Calculate 2D and 3D vectors with exact step-by-step solutions: addition, magnitude, dot product, cross product, triple product, angle, orthogonality, and area/volume with an interactive 3D visualizer.
3D Isometrische Vektor-Visualisierung
Nicht orthogonal
1.2373 rad
Dreieck: 4.3302 FE
Abstand d(a,b) = 3.7417 LE
Normalenvektor senkrecht auf a und b mit Länge |a × b| = 8.6603.
Die Vektoren spannen einen echten 3D-Körper auf (linear unabhängig).
Vollständiger mathematischer Lösungsweg & Formeln
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Physical Forces in 3D Space Without Coordinate Clutter
Forces, velocities, and electromagnetic fields possess both magnitude and spatial direction. In the early 19th century, physicists had to write cumbersome scalar equations for each individual coordinate axis (x, y, z).
2. Vector Calculus, Dot & Cross Products
Gibbs and Heaviside extracted modern vector analysis from Hamilton's quaternions. Defining vectors with dot and cross products made 3D mechanical work, angles, and torque calculations extraordinarily streamlined.
3. Aerodynamics, Space Flight & Physics Game Engines
Aircraft flight dynamics, rocket trajectory simulation, and rigid-body collision physics in video games rely entirely on vector math.
Real-World Applications of Vector Algebra
Vectors provide the mathematical language for physics, forces, computer graphics, robotics, and flight navigation.
Flight Heading & Velocity Vectors
Navigation computers vectorially combine aircraft airspeed and heading with wind velocity vectors to determine the actual ground track.
Airspeed vector (250, 50, 10) + Wind vector (-30, 10, 0) = Resulting ground track vector (220, 60, 10) km/h.
Shading, Lighting & Surface Normals
Graphics engines use the cross product to compute surface normal vectors required for realistic ray tracing and lighting reflections.
Polygon edges u and v yield the surface normal n = u × v used in Lambertian reflectance models.
Force Decomposition in Truss Bridges
Tension and compression forces across truss nodes are resolved into orthogonal vector components ensuring static equilibrium.
Bridge cable tension is decomposed into horizontal and vertical components: F = F_x + F_y + F_z.
Mathematical Guide to Vector Calculus
A vector represents a quantity possessing both magnitude and direction in Euclidean space. Vectors are fundamental to linear algebra and analytic geometry.
Vector Addition & Magnitude
Addition is computed component-wise: a + b = (a1+b1, a2+b2, a3+b3). Vector magnitude is calculated via the Euclidean norm: |a| = √(a1² + a2² + a3²).
Dot Product & Orthogonality
The dot product a · b = a1b1 + a2b2 + a3b3 measures parallelism. If a · b = 0, the vectors are orthogonal (perpendicular at 90°).
Cross Product (Vector Product)
The cross product a × b produces a 3D vector perpendicular to both input vectors. Its magnitude equals the area of the spanned parallelogram.
Scalar Triple Product & Volume
The scalar triple product [a, b, c] = (a × b) · c determines the volume of the parallelepiped. If [a, b, c] = 0, the vectors are coplanar.
Frequently Asked Questions (FAQ) on Vectors
What is the difference between dot product and cross product?▼
The dot product produces a single scalar number used for angles. The cross product produces a new 3D vector perpendicular to both vectors.
When are two vectors collinear (parallel)?▼
Two vectors are collinear if one is a scalar multiple of the other (b = k · a) or their cross product equals the zero vector.
How do you calculate the angle between two vectors?▼
Using the formula cos(φ) = (a · b) / (|a| · |b|). The angle φ is calculated via the arccosine function.
What does coplanar mean?▼
Vectors are coplanar if they all lie within the same geometric plane, which occurs precisely when their scalar triple product is 0.