Words as Vectors

Vector Differences Are Exact

Before dot products, the displayed vectors can be subtracted coordinate by coordinate. This lesson shows that each difference row is exact arithmetic on the assigned toy coordinates, not a new hidden source.

highlighted = computed this step

Subtract coordinates

Vector subtraction is coordinate arithmetic. Use the same displayed rows: cat=(3,1), kitten=(2,1), and car=(1,3). Each difference subtracts the right vector from the left vector.

(x1,x2)−(y1,y2)=(x1−y1,x2−y2)(x_1,x_2)-(y_1,y_2)=(x_1-y_1,x_2-y_2)
Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.exact coordinate differencescat=(3,1); kitten=(2,1); car=(1,3)every row subtracts right coordinates from left coordinatespaircoordinate subtractiondifferencesquared totalcat-kitten(3-2,1-1)(1,0)1^2+0^2=1cat-car(3-1,1-3)(2,-2)2^2+(-2)^2=8kitten-car(2-1,1-3)(1,-2)1^2+(-2)^2=5assigned toy coordinates only; subtraction stays exact

Two differences from cat

Cat minus kitten is (3 minus 2,1 minus 1) = (1,0). Cat minus car is (3 minus 1,1 minus 3) = (2,-2).

cat−kitten=(1,0),cat−car=(2,−2)\text{cat}-\text{kitten}=(1,0),\quad \text{cat}-\text{car}=(2,-2)
Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.exact coordinate differencescat=(3,1); kitten=(2,1); car=(1,3)every row subtracts right coordinates from left coordinatespaircoordinate subtractiondifferencesquared totalcat-kitten(3-2,1-1)(1,0)1^2+0^2=1cat-car(3-1,1-3)(2,-2)2^2+(-2)^2=8kitten-car(2-1,1-3)(1,-2)1^2+(-2)^2=5assigned toy coordinates only; subtraction stays exact

Kitten minus car

Kitten minus car is (2 minus 1,1 minus 3) = (1,-2). If squared totals are shown, they are also exact: 1, 8, and 5. No square-root distance is introduced.

kitten−car=(1,−2)\text{kitten}-\text{car}=(1,-2)
Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.exact coordinate differencescat=(3,1); kitten=(2,1); car=(1,3)every row subtracts right coordinates from left coordinatespaircoordinate subtractiondifferencesquared totalcat-kitten(3-2,1-1)(1,0)1^2+0^2=1cat-car(3-1,1-3)(2,-2)2^2+(-2)^2=8kitten-car(2-1,1-3)(1,-2)1^2+(-2)^2=5assigned toy coordinates only; subtraction stays exact

Boundary

These are assigned toy coordinates only. The subtraction rows do not claim learned embeddings, meaning, semantics, understanding, model quality, or generalization. They only show that later vector arithmetic comes from the same displayed integers.

assigned toy vectors; exact coordinate arithmetic\text{assigned toy vectors; exact coordinate arithmetic}
Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.Vector Differences Are ExactCoordinate subtractions are recomputed from displayed vectors.exact coordinate differencescat=(3,1); kitten=(2,1); car=(1,3)every row subtracts right coordinates from left coordinatespaircoordinate subtractiondifferencesquared totalcat-kitten(3-2,1-1)(1,0)1^2+0^2=1cat-car(3-1,1-3)(2,-2)2^2+(-2)^2=8kitten-car(2-1,1-3)(1,-2)1^2+(-2)^2=5assigned toy coordinates only; subtraction stays exact