About this experiment
Build vectors by editing their components or dragging their tips. Compare head-to-tail addition, differences, scalar products, projections and the right-handed normal of a cross product.
Learning goals
- Relate coordinate components to a spatial vector.
- Distinguish dot products, cross products and projections.
- Recognise the undefined direction of a zero reference vector.
How to use it
- 1Drag the A tip along x and the B tip along z and compare the changing components.
- 2Inspect the parallelogram for vector addition.
- 3Make the vectors perpendicular and check the dot product and angle.
- 4Choose projection, then set B to zero and interpret the undefined result.
The model
Assumptions
- All results use Euclidean three-dimensional coordinates and exact component formulas up to floating-point rounding.
- The cross-product arrow displays its unit direction, not its raw magnitude. Numerical components and magnitude retain the full cross product.
- The two drag tips control one component each; all six components remain editable. Camera orbit does not change the vectors.
Try this in class
- Q1Why can the dot product be negative while the angle remains between zero and 180 degrees?
- Q2Why is projection onto a zero vector undefined?
Frequently asked questions
Why does the cross-product arrow stay short when its magnitude grows?
Its displayed length is one unit so its direction remains visible. The full magnitude is reported separately.
Does a zero dot product always mean an angle of 90 degrees?
Only when both vectors are nonzero. A zero vector has zero dot product with every vector but no defined direction or angle.