Vector Addition Calculator

Vector addition combines two vectors by adding their corresponding components. For 2D vectors A = (a1, a2) and B = (b1, b2), the sum is A + B = (a1 + b1, a2 + b2). The result is a new vector. Geometrically, vector addition is performed by placing vectors head-to-tail.

Vector subtraction A - B is equivalent to adding the negative: A - B = A + (-B). Component-wise, (a1, a2) - (b1, b2) = (a1 - b1, a2 - b2). Geometrically, it represents the vector from B to A.

Scalar multiplication multiplies a vector by a real number (scalar). If A = (a1, a2) and k is a scalar, then kA = (k*a1, k*a2). This scales the vector's magnitude. If k > 1, the vector gets longer. If 0 < k < 1, it gets shorter. If k < 0, it points in the opposite direction.

The magnitude (or length) of a vector A = (a1, a2) is |A| = sqrt(a1^2 + a2^2). For 3D vectors (a1, a2, a3), it is |A| = sqrt(a1^2 + a2^2 + a3^2). This represents the distance from the origin to the point (a1, a2) or (a1, a2, a3).

Vector addition is used in physics to combine forces and velocities, in computer graphics for transformations and positioning, in navigation for combining displacements, and in engineering for analyzing complex systems with multiple components.