Vector Definition

Pre-Calculus · Axiom Academy

A vector is more than a number — it carries both a size and a direction. See what that means as an arrow. 1. An Arrow With Length and Heading Picture an arrow drawn from a starting point. Two facts describe it: how long it is (its magnitude) and which way it points (its direction). Watch the arrow grow — the length read-out is the magnitude, and the sweeping angle is the direction. Magnitude — the arrow's length Direction — the angle from the +x axis 2. Components Are Run, Then Rise Every vector has an initial point P_1 where it starts and a terminal point P_2 where it ends. To get from tail to tip, go across by and then up by . Those two numbers are the vector's components . tells you how far across to move, and its sign tells you left or right. tells you how far up to move, and its sign tells you up or down. From P_1(1, 1) to P_2(7, 5) , the run is 7 - 1 = 6 and the rise is 5 - 1 = 4 , so the vector is — exactly what the two legs in the animation spell out. Here is the surprise: a vector does not remember where it started. Slide the arrow to a new spot without turning or stretching it, and it is the same vector . Watch one arrow translate across the plane — every copy has the identical components . This is why we can talk about a vector by its components alone — once you know "3 across, 2 up," the starting point is just a choice of where to draw it. The copy drawn from the origin (0,0) . It pins down a single point's location — answers "where is this point?"

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