Unit 9: Dot products and duality
About this unit
The dot product multiplies matching entries and adds. Geometrically it is a shadow length times a length, so its sign says same side, perpendicular or opposite. And every linear map from vectors to numbers is a dot product with one fixed vector.
By the end, you can: Compute dot products, read them as projections with a sign, and see every linear map from the plane to the number line as a dot product with one vector.
Modules
5 modulesLearn
- Multiply, add, and a shadowArticle · The dot product multiplies matching entries and adds; geometrically it is the length of one vector's shadow on the other times the other's length, and its sign says whether they point the same way.Not started
- Make the dot product zeroExperiment · Drag u until u · v = 0 and find that it happens exactly when u is perpendicular to v.Not started
- Vectors that are also mapsArticle · A linear map from the plane to the number line is a 1×2 matrix, and applying it is the same as a dot product with the vector of its entries: vectors and such maps pair up one to one (duality).Not started
- One number from two vectorsReview · The formula, the projection picture, the sign, symmetry and duality.Not started