DotProduktWert

DotProduktWert is a scalar value that results from the dot product of two vectors of the same dimension. It is a fundamental operation in linear algebra and is used to quantify the degree of alignment between two vectors.

Definition and formula

Let u = (u1, u2, ..., un) and v = (v1, v2, ..., vn) be vectors in R^n. The DotProduktWert, often written as dot(u, v) or u · v, is the sum of the products of corresponding components:

DotProduktWert = sum_{i=1}^n ui vi.

In two dimensions this reduces to u1 v1 + u2 v2.

Geometric interpretation

The dot product relates to the angle theta between the vectors by DotProduktWert = ||u|| ||v|| cos(theta), where ||u|| and ||v|| are the Euclidean norms of u and v. When the vectors are identical, DotProduktWert = ||u||^2. When the vectors are orthogonal, DotProduktWert = 0.

Properties

DotProduktWert is commutative: u · v = v · u. It is bilinear and distributes over vector addition: (u + w) · v = u · v + w · v. It is non-negative on the same vector and scales with magnitude: k(u · v) = (ku) · v = u · (kv) for any scalar k.

Applications and computation

The dot product is used to measure similarity, compute projections, and determine orthogonality. It underpins cosine similarity, linear regression, and many graphics and physics computations. In programming, DotProduktWert is computed via loops or optimized linear algebra libraries as dot(u, v) = sum ui vi. Edge cases include dimension mismatch errors and DotProduktWert = 0 for orthogonal vectors; the zero vector requires special attention in normalization contexts.