From Points to Vectors

Linear Algebra (Matrices) · Axiom Academy

A point on the plane is a place. Turn it into a vector — an arrow from the origin — and suddenly you can add and scale . That's where the algebra in linear algebra comes from. A point is a place. A vector is a move. You already plot points: (3,2) means "3 right, 2 up." A vector takes that same pair and reads it as an arrow from the origin — a length and a direction pointing out to the spot. The pair (x,y) doesn't change; what you can do with it does. Watch each scattered point sprout an arrow from the origin — the point becomes its position vector . Then two of those arrows chain tip-to-tail to make their sum, and one gets stretched . Points can't be added or stretched; vectors can — and that single difference is the whole game. Same coordinates, new powers: once a point is an arrow, it has a length and a direction you can combine with others. Drop a point — watch its vector appear Drag the point anywhere on the grid. The arrow from the origin out to it is its position vector. Read its components and its length change as you move — the point and its vector are two views of the same pair of numbers. The point is where ; the vector is how to get there from the origin — length and direction, baked into the same . Drag the tips of and . Slide so it starts at the tip of : where it lands is , the arrow straight from the origin. Adding vectors just adds components — — something you simply cannot do to two points.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.