Perform 2D vector addition, subtraction, and scalar multiplication. Enter components for u and v, choose an operation, and see component-by-component arithmetic, magnitudes, and a live canvas diagram illustrating the parallelogram law.
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Vector Addition — Parallelogram Law & Head-to-Tail
u + v = ⟨a+c, b+d⟩
Component method: add the x-components together and the y-components together — that's it. No angles required.
Parallelogram law: place both vectors with their tails at the origin. The resultant u + v is the diagonal of the parallelogram formed by u and v. This is the violet arrow in the diagram.
Head-to-tail method: draw u, then draw v starting from the head of u. The resultant is the arrow from the very beginning of u to the very end of v. Both methods give the same answer — they are two geometric interpretations of the same arithmetic.
Scalar multiplication stretches (|k| > 1), shrinks (|k| < 1), and/or reverses (k < 0) a vector without changing its direction: k·u = ⟨ka, kb⟩.
Magnitude vs Direction — What Adding Vectors Means Geometrically
|u| = √(a² + b²) |u + v| ≤ |u| + |v|
The magnitude (length) of a vector is the distance from its tail to its head — computed by the Pythagorean theorem on its components. Magnitude tells you "how far," while direction (the angle) tells you "which way."
Adding two vectors combines two displacements. If u = "walk 3 blocks east" and v = "walk 2 blocks north," then u + v = "walk to the position you'd reach doing both." The resultant arrow is the shortcut path.
The Triangle Inequality guarantees |u + v| ≤ |u| + |v|: you can never get a shortcut longer than taking both trips separately. Equality holds only when both vectors point in exactly the same direction.
Vector subtraction u − v = u + (−v). Geometrically, flip v to get −v, then add. The diagram shows −v as the same length but pointing opposite to v, then adds it to u.
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