Vector Fundamentals

Calculus 3 · Axiom Academy

A vector is an arrow — magnitude and direction packed into its components , and every operation is just something you do to that arrow. 1. A Vector Is a Directed Arrow Draw an arrow from the origin to a point. Its direction is where it points; its length is how far it goes. We record it by how far it reaches across and up — those two numbers are its components . The arrow from the origin to the point (a, b) Its components: run a across, rise b up 2. Magnitude Is the Arrow's Length The magnitude is simply how long the arrow is. The run and the rise are the two legs of a right triangle, and the arrow is its hypotenuse — so the Pythagorean theorem measures it directly. The run a and rise b meet at a right angle — the sides of the triangle under the arrow. The arrow spans the corner to the tip. Its length is . To add two vectors, slide the second so its tail starts where the first one's tip ended. The sum is the single arrow from the very start to the very end — the net displacement of taking one trip, then the other. Componentwise, this is just addition: line the vectors up and add matching entries. Multiplying a vector by a scalar k stretches or shrinks the arrow along its own line — the direction is unchanged, only the length scales. Every component is multiplied by k . A vector is an arrow carrying magnitude and direction, recorded by its components — and every operation is a move on that arrow.

This is the written version of the interactive lesson above. See the full Calculus 3 course.