Unit Vectors and Direction

Calculus 3 · Axiom Academy

LESSON Unit Vectors and Direction Every vector splits into a size and a direction — and the unit vector is that direction, distilled to length 1. 1. Strip a Vector Down to Its Direction Take . Its length is . To keep only the direction , divide every component by that length. The tip slides straight back along the same line until it rests on the unit circle — that landing point is the unit vector . 2. The Ready-Made Unit Vectors: i, j, k Three unit vectors come built in — one pointing straight along each axis. They are the standard basis . Each has length 1, and each points purely along one direction, so any vector can be read as a combination of them: . Points along the positive x-axis. Length 1. Points along the positive y-axis. Length 1. Points along the positive z-axis. Length 1. The three meet at right angles and frame all of 3D space. are the yardsticks of direction. Writing just says " a steps east, b steps up, c steps out" — each step measured by a unit vector. 3. Rebuild Any Vector: Size Times Direction The move runs both ways. Start from the unit vector on the circle and scale it back out by the number : it grows straight along its own line into . So every vector factors cleanly into a size and a direction . The scalar carries all of the length — one number, no direction. The unit vector carries all of the direction — length locked at 1. Scale by any positive number to get a vector that points the same way but is longer or shorter.

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