Magnitude and Direction

Pre-Calculus · Axiom Academy

When a number has to point somewhere A number, all by itself, can leave you stranded You're lost in a city and ask for directions. The answer comes back: "Go 3 blocks." You start walking — then stop. Three blocks which way? North, south, east, west? The number 3 tells you how far , but not which direction , and that missing half can leave you completely stranded. A temperature is just a number. It points nowhere; turn around and it's still 72°. A wind has a speed and a heading. "15 mph northeast" is nothing like "15 mph south." Watch what "3" does without a direction: it could send you anywhere on the dashed circle. A vector fixes that — it picks one arrow, carrying both the size (its length) and the direction (where it points) at once. Size and direction, bundled into one arrow — that arrow is a vector. Build a vector: pick a length, pick a direction Drag the magnitude r and the direction — or grab the arrow's tip and swing it. The arrow always points degrees from the positive x -axis and reaches r units out. The dashed legs show how far it goes across and up : its components . "5 units at 45°" and "3.54 across, 3.54 up" describe the very same arrow. Grab the tip and drop it anywhere on the grid. Every spot has two equally good descriptions of the same arrow: its components (run and rise) and its magnitude and direction . The dashed right triangle is the bridge — and is just its hypotenuse. Same arrow, two languages: is exactly "5 units at about 36.9°."

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