Span Builder

Each challenge checks itself. Work on this keeps it in view above the controls.

Choose two or three vectors in the plane or in space, drive the coefficients with sliders to hit a target vector, and see the span: a line, a plane or everything.

How to use

• A linear combination is w = c₁v₁ + c₂v₂ (+ c₃v₃). Move the coefficient sliders (steps of 0.5) or type any number in the boxes next to them.
• The picture draws the combination tip to tail: first c₁v₁ from the origin, then c₂v₂ from its tip, and so on. The black arrow w goes from the origin to the end of the chain. Try to land it on the star, the target b.
• The span is the set of all combinations. If the vectors are independent in the plane, the span is the whole plane. If they are dependent (all on one line), the span collapses to that line, drawn as a wide band.
• The Facts list states independence or a dependency relation, and whether b is in the span.
• Keyboard: Tab to a slider and use the arrow keys (Page Up/Page Down for bigger steps). The status box is read aloud by screen readers.
• Patterns tell the vectors apart: v₁ solid, v₂ dashed, v₃ dotted; w is a thin black arrow; b is a star.

Accessibility: every control is a standard form control you can reach with Tab and change with the arrow keys; the status box is announced by screen readers; the picture has a text description (press Describe) that updates as you work; animations are skipped when your system asks for reduced motion.