Vector Product Calculator & Tutorial
An interactive guide to understanding and calculating the Dot Product, Cross Product, and angle between two 3D vectors, complete with a live 3D visualizer.
What is a Vector?
In physics and math, a vector is a quantity that has both magnitude (length or size) and direction. Think of it as an arrow pointing from a starting point to an ending point. This is different from a scalar, which only has magnitude (like temperature or speed). Vectors are essential for describing forces, velocity, and displacement in 3D space.
A Vector A
B Vector B
The 'Aha!' Moment: An Engineering Student's Story
Meet Alex, a first-year engineering student. In physics class, Alex is struggling with the concept of torque. The problem is: "A force vector F = ⟨4, 5, 0⟩ is applied to a wrench at a position vector r = ⟨0.2, 0.1, 0⟩ from the pivot point. What is the torque vector, and what is its magnitude?"
Remembering that torque is the cross product of r and F (τ = r x F), Alex opens the Vector Calculator. By entering Vector A as r = ⟨0.2, 0.1, 0⟩ and Vector B as F = ⟨4, 5, 0⟩, Alex instantly gets the torque vector and its magnitude.
The 3D visualization makes it click: the new green vector (torque) is perpendicular to both the wrench (r) and the force (F), showing exactly how the rotational force is generated. The abstract formula becomes a tangible, visual concept.
Geometric Interpretation of Vector Products
The Dot Product: A Measure of Alignment
The dot product (A · B) tells you how much two vectors point in the same direction. It results in a single number (a scalar).
• If A · B > 0, the vectors point in a similar direction (angle < 90°).
• If A · B = 0, the vectors are perfectly perpendicular (angle = 90°).
• If A · B < 0, the vectors point in opposite directions (angle > 90°).
Our visualization shows this as the projection of vector A onto vector B (the magenta arrow).
The Cross Product: The Perpendicular Vector
The cross product (A x B) creates a new vector that is perpendicular to the plane formed by the original two vectors.
• Its direction is determined by the "Right-Hand Rule".
• Its magnitude `|A x B|` is equal to the area of the parallelogram formed by vectors A and B. This is why if A and B are parallel, their cross product is the zero vector (area is zero).
Our visualization shows this as the green arrow.
Real-World Applications
Physics: Work and Torque
The dot product is used to calculate the work done by a force, which is `Work = F · d`. The cross product is essential for calculating torque (`τ = r x F`), which measures rotational force, and for understanding magnetic fields and angular momentum.
Computer Graphics & Game Dev
The dot product is used to determine lighting angles on 3D models (how much a surface is facing a light source). The cross product is used to calculate surface normals (the direction a surface is facing), which is critical for lighting, reflections, and "back-face culling" to optimize rendering performance.
Engineering & Mechanics
In structural engineering, vector products are used to analyze forces acting on beams and trusses. The dot product can determine the component of a force in a specific direction, while the cross product helps in analyzing the moments and rotational effects on structures.
Formulas in Detail
Dot Product
Cross Product
Vector Magnitude
Angle Between Vectors
Frequently Asked Questions
What does the 3D visualization show?
The interactive 3D plot shows your input vectors (A in blue, B in red), the resulting cross product vector (C in green), and the projection of A onto B (magenta). You can click and drag to rotate the view, and use the scroll wheel to zoom. The orange arc visually represents the angle between A and B.
What is the "Right-Hand Rule"?
The direction of the cross product vector (A x B) follows the right-hand rule. If you point your index finger in the direction of vector A and your middle finger in the direction of vector B, your thumb will point in the direction of the cross product.
Why is the dot product a scalar?
The dot product is a projection of one vector onto another, scaled by the magnitude of the second vector. This operation fundamentally results in a measure of "how much" they align, which is a scalar quantity (a single number), not a directional one.