RECALLWEAVE / VECTOR GEOMETRY

See what a projection keeps.

Change a vector and the direction of a line through the origin. The projection stays on the line; the residual shows what is left over. Try the course example, then compare a longer direction for the same line.

Choose the geometry

Coordinates use perpendicular axes with equal units. Enter values from −6 to 6 in steps of 0.25.

Vector b
Line direction a

Try a relationship

One vector, two perpendicular parts

Vector projection on a line through the origin The current vector, its projection, and its residual are drawn on the same coordinate axes.
b: original vectorp: projectionr: residual

The gray line continues in both directions. The short gray arrow marks a; changing its length alone keeps the line in the same place.

Original vector b
Projection p
Residual r = b − p
Dot product b · a
Squared direction length a · a
Residual dot direction r · a

Shown values are rounded to three decimal places. In the exact geometry, the residual's dot product with the line direction is zero.

Check both parts

The projection must lie on the line, and b minus that point must be perpendicular to the line. A perpendicular arrow pointing the other way does not reconstruct b. For a nonzero direction a, the coefficient is (b · a) / (a · a).

Connect it to the course

The twelve-question lesson continues from displacements and unit directions to dot products, projection, matrix actions and operation order. Download the local JSON, choose it in RecallWeave's deck importer, review it, and explicitly start the lesson.