This sandbox was generated by Claude (Anthropic) for the Linear Algebra 101 course materials, and is built for Module 1. Set two vectors u and v by dragging their tips on the plane or typing their components, and watch everything the module defines update at once: the sum (the diagonal of the parallelogram), a linear combination a u + b v, the dot product, the angle between them, the determinant, and the projection of u onto v. Every drawn point is computed — the tip of the sum, the foot of the projection and the span line all come from the vector arithmetic, never sketched.
The determinant decides the pair. det(u, v) = u₁v₂ − u₂v₁ is the signed area of the parallelogram u and v build. When it is non-zero the vectors are independent and their span is the whole plane; when it is zero they are dependent and their span collapses to a line (or a point, if both are zero). The page prints the determinant, the independence verdict and the span dimension together, and draws the span as a line when the parallelogram has flattened to zero area.
Notation follows the course: column vectors, the dot product written u . v, the norm |u|, the angle from cos θ = (u . v)/(|u| |v|), and the projection projₖ(u) = ((u . v)/(v . v)) v.
What is chosen rather than computed: the default vectors, the preset pairs (each labelled where it is selected), and the extent of the drawn grid. Nothing on this page is a measurement of anything; it is arithmetic, and the arithmetic is done rather than imitated.
Course demo — linked from the Module 1 lesson deck; the page itself is English‑only for now. Two vectors in the plane carry a surprising amount of structure. Drag them around and watch it all follow. One thing to take away: the determinant u₁v₂ − u₂v₁ is zero exactly when the two vectors line up — and that is the same moment their span stops being the whole plane and becomes a single line.