Vector Operations

Pre-Calculus · Axiom Academy

Adding, subtracting, and scaling vectors — and why the component algebra always matches the picture. To add and , slide over so its tail starts at 's tip . The single arrow from 's start to 's new end is the sum . Algebraically, you just add matching components — and the arrow lands at exactly that point. 2. Subtraction: The Connecting Arrow Draw and from the same starting point. Then is the arrow that points from the tip of to the tip of — the displacement that takes you from where ends to where ends. As components, you subtract term by term. Place both and with their tails at the origin. The difference runs from 's tip to 's tip, the short way across. It's the arrow you'd add to to reach : . Equivalently, — flip and add tip-to-tail. With and , the difference is — it points right and down, exactly the arrow from 's tip back to 's tip. 3. Scalar Multiplication: Stretch and Flip Multiplying a vector by a number k scales each component by k . Geometrically the arrow keeps its line but changes length — and if k is negative, it reverses direction . Watch k run from -1 up to 2 : the arrow shrinks to a point at k=0 , then grows the other way. points the same way but is twice as long. keeps the direction, half the length. has the same length but points the opposite way. You've seen all three vector operations as both component formulas and moving arrows — the algebra and the geometry are one and the same. Scroll up to revisit any step.

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