Vector Fundamentals

Calculus 2 · Axiom Academy

Magnitude and direction in one object — and the three operations that build everything else: adding them tip-to-tail, scaling them, and measuring their length. A vector is drawn as an arrow : its length is the magnitude and its tilt is the direction. Crucially, position doesn't matter — slide the arrow anywhere without rotating or resizing it and it's still the same vector. The animation builds one arrow , then copies it to a new spot to show the copy is identical. Drop the arrow's tail at the origin. Then the tip lands at a point, and its coordinates are the components: how far right ( a ) and how far up ( b ). The animation walks the run, then the rise, for — the two legs that assemble the arrow. Here and are the unit vectors along the axes, so and name the very same arrow. The components are the two legs of a right triangle; the vector itself is the hypotenuse . So its length is just the Pythagorean theorem. The animation grows the legs of , snaps in the right angle, then draws the hypotenuse and resolves it to a clean . To add geometrically, slide the second arrow so its tail sits on the first arrow's tip . The sum runs from the very first tail to the very last tip. The animation does exactly this with and , then draws the resultant — which lands on , the component-wise sum. Because you can lay the two arrows in either order and still reach the same corner, addition is commutative: . That same corner is the diagonal of the parallelogram the two vectors span.

This is the written version of the interactive lesson above. See the full Calculus 2 course.