Vektorsumman

Vektorsumman, or the vector sum, is the result of adding two or more vectors in a vector space. For vectors u and v in R^n, with components u = (u1, ..., un) and v = (v1, ..., vn), their sum is u + v = (u1 + v1, ..., un + vn). The operation is defined by the axioms of a vector space and is both associative and commutative. The additive identity is the zero vector, and the additive inverse of a vector u is -u, since u + (-u) = 0.

Geometric interpretation often uses the parallelogram rule or the head-to-tail method. The magnitude of the sum is not generally equal to the sum of magnitudes, and it satisfies the triangle inequality: ||u + v|| ≤ ||u|| + ||v||, with equality when u and v point in the same direction. The direction of the sum depends on the relative directions and lengths of the original vectors.

In coordinates, addition is performed component-wise, and the result remains in the same space. For a scalar a, the distributive properties hold: a(u + v) = au + av and (a + b)u = au + bu.

Common applications of vektorsumman appear across science and engineering. It is used to compute resultant forces in physics, total displacement in navigation, and color or lighting calculations in computer graphics. It also underpins many algorithms in data processing and numerical methods.

Example: if u = (2, 3) and v = (-1, 4), then u + v = (1, 7), with magnitude ||u + v|| = sqrt(50) ≈ 7.07. The concept is foundational in linear algebra and vector analysis.