This simulator builds two free Euclidean vectors A and B in 3D, then translates B to A's tip to construct the head-to-tail sum (or difference) and the completing parallelogram. Adjust each vector's x, y and z components independently, switch between addition and subtraction, and animate the translation to see that moving an arrow's position never changes its underlying components.
• A real-time 3D view of vector A anchored at the origin, a translated second vector (B, or −B during subtraction), the resultant vector and its endpoint, a completing parallelogram guide, and the x/y/z reference frame with x and y in the ground plane and z vertical — with home view, focus-selected-part, auto-rotate, expand and toggleable labels. • Eight live controls: A's x, y and z components, B's x, y and z components (each −2 to 2), an operation selector (A + B or A − B), and a head-to-tail placement slider from 0 (common tail) to 1 (translated to A's tip). • Four actions: build head-to-tail (animate), complete construction, swap A and B, and pause placement. • Seven live metrics: result x, y and z components, result length, |A|, |B| and the angle between A and B. • A Curves & measurements tab plotting result x and y, the full model equations and snapshot measurements. • An Experiments tab with four guided scenarios (right-angle addition, cancellation, subtracting a vector, swapping operands), a model-verification bench of independent automated checks, and a timestamped event log with a copyable trial report. • A Learn & assess tab with guided lessons, a two-question knowledge-check quiz and a written scope/reference statement.
A free vector is defined entirely by its components — Ax, Ay, Az — not by where it happens to be drawn. Sliding the placement control from 0 to 1 moves the second arrow's tail from the shared origin to A's tip, but at every point along that animation the resultant components Rx = Ax ± Bx, Ry = Ay ± By and Rz = Az ± Bz stay exactly what they were before the translation started. At an intermediate placement value, the moving arrow's tip is not yet the resultant's endpoint — only once the translation completes does the tip land on A + B (or A − B).
Subtraction is constructed by first reversing every component of B to get −B, then chaining it head-to-tail after A exactly as with addition — which is why the simulator's violet arrow represents −B specifically during a subtraction operation, not B itself.
The equations panel shows R = A + B (or A − B) resolved componentwise, |R| = √(Rx² + Ry² + Rz²), and cos θ = (A·B)/(|A||B|) for nonzero vectors. When A and B are exact opposites, their sum is the zero vector — which has no unique resultant direction or defined angle, since a zero-length vector carries no orientation information.
Addition is commutative (A + B always equals B + A, as the swap-operands experiment confirms), but subtraction generally is not: B − A is the negative of A − B, so swapping the two operands in a subtraction reverses the resultant's direction. This is a free-vector model — the placement animation is a display convenience for visualizing the sum, and it never alters the underlying mathematical result.
No, as long as its length and direction stay the same. A free vector is determined entirely by its components, not by where it is drawn — translating the second arrow to build the head-to-tail chain repositions it visually but leaves its x, y and z components, and therefore the computed resultant, completely unchanged.
No. Addition is commutative — A + B always equals B + A — but subtraction generally is not. B − A is the exact negative of A − B, so swapping which vector comes first in a subtraction reverses the direction of the resultant, which you can confirm directly in the swap-operands experiment.
Their sum is the zero vector, and the simulator has no unique resultant direction or angle to display in that case — a zero-length vector carries no defined orientation. The cancellation experiment demonstrates this by combining equal and opposite vectors and showing the resultant collapse to zero.
This is a free Euclidean vector model: vectors are defined only by their components, with no notion of a bound application point, force line of action or physical units attached. The head-to-tail placement animation is a display construction for visualizing the sum geometrically — it does not represent a physical motion of anything in the world, and it never changes the underlying resultant computed from the components.