Unit Vectors

Pre-Calculus · Axiom Academy

A vector of length 1 is pure direction — the answer to "which way?" with the "how far?" stripped away. 1. A Unit Vector Is Pure Direction A unit vector is a vector whose magnitude (length) is exactly 1. Watch the arrow below sweep through every direction: the tip rides the unit circle , and no matter where it points, its length never budges from 1. Direction changes; length stays pinned at one. a unit vector at angle — its tip is a point on the unit circle 2. The Standard Unit Vectors and Two unit vectors do special work: points one step in the x -direction, and points one step in the y -direction. Any vector is built by laying down some copies of and some copies of , tip to tail. Watch run across, then climb up — together they land on . Go a steps along — that covers the horizontal component. Then b steps along — that covers the vertical component. is the same vector as — two names for one arrow. The coefficients a and b are the coordinates of the tip. means "3 right, 2 up." So , and its tip sits at the point (3, 2) . 3. Normalizing: Shrinking to Length 1 To get the unit vector pointing the same way as any vector , divide by its own length . That rescales the arrow — without turning it — until its tip lands exactly on the unit circle. This is called normalization . Watch (length 5) collapse straight down its own line onto length 1. , length . Long arrow, some direction. , length 1 . Same direction, unit length.

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