skalaarkorrutamise

Skalaarkorrutamise, also known as the dot product or scalar product, is a fundamental operation in linear algebra and vector calculus. It takes two vectors of the same dimension and returns a single scalar value. For two vectors u and v, each with n components, their scalar product is calculated by multiplying corresponding components and summing the results. Mathematically, if u = [u1, u2, ..., un] and v = [v1, v2, ..., vn], then the scalar product is u · v = u1*v1 + u2*v2 + ... + un*vn.

The geometric interpretation of the scalar product is particularly significant. It relates to the magnitudes of the two vectors and the angle between them. Specifically, u · v = ||u|| ||v|| cos(θ), where ||u|| and ||v|| represent the magnitudes (or lengths) of vectors u and v, respectively, and θ is the angle between them. This relationship makes the scalar product useful for determining if vectors are orthogonal (perpendicular), as their dot product will be zero if cos(θ) = 0.

Applications of the scalar product are widespread. It's used in physics to calculate work done by a force, in computer graphics for lighting calculations and surface normal projections, and in machine learning for similarity measures between data points represented as vectors. The scalar product provides a way to quantify the alignment or projection of one vector onto another.