Discover vectors as arrows, not just lists of numbers. Explore addition, scaling, and what it means to span a space.
A vector is an arrow from the origin to a point, and its numbers just say where the tip lands. Drag, stretch, and combine arrows below to see what that buys you.
Drag the blue and green tips and watch the sum close the parallelogram. Walking the two arrows tip to tail always lands on the same point.
Drag the blue and green vector tips. Notice how the sum always completes the parallelogram.
Before diving into matrices, let's build intuition for vectors. A vector isn't just a list of numbers—it's an arrow pointing from the origin to some point in space. The numbers simply tell us where the tip lands.
This geometric view is the key to understanding all of linear algebra. Every operation we'll learn has a beautiful visual meaning.
To add two vectors, place the tail of the second at the tip of the first. The sum is the vector from the original origin to the new tip—forming a parallelogram.
Slide the scalar and watch the arrow stretch, shrink, and flip. Positive scalars keep its direction; negative scalars reverse it.
Positive scalars: stretch or compress the vector
Negative scalars: flip direction and scale
Zero: collapses the vector to a point
Dial the two coefficients and watch where the scaled-and-added vector lands. Try to reach a point you pick in advance.
A linear combination scales each basis vector and adds them together. Every point in the plane can be reached by some combination of v₁ and v₂.
A linear combination takes multiple vectors, scales each by some coefficient, and adds the results. This is the fundamental operation of linear algebra.
Drag the two vectors and watch the reachable region change. Make them parallel and see the plane shrink to a line.
Two non-parallel vectors span the entire 2D plane
The span of a set of vectors is all possible linear combinations. Drag the vectors to see how the span changes.
The span of a set of vectors is all points reachable via linear combinations. One vector spans a line. Two non-parallel vectors span a plane. What happens if they're parallel?
Span is all points reachable; non-parallel vectors span more.
Drag to rotate the view and see the arrows in space. Three independent vectors reach every point in 3D.
Three independent vectors span all of 3D space. Drag to rotate the view, scroll to zoom. Any point in space can be reached by some linear combination of v₁, v₂, and v₃.
Spot a mistake or have an idea?