The dot product turns two arrows into one number
Add the products of matching coordinates, or multiply lengths by the cosine of the angle between them—in Cartesian coordinates those two recipes agree. That single scalar rebuilds length and angle in the modern formulation of Euclidean geometry.
Algebraically the dot product of two equal-length number sequences sums the products of corresponding entries. Geometrically the scalar product of two Euclidean vectors is the product of their magnitudes and the cosine of the included angle. Fix a Cartesian frame and the two stories coincide. The name "dot product" comes from the ⋅ notation; "scalar product" stresses that the output is a scalar, not a vector like the three-dimensional cross product. "Inner product" is the broader abstract label of which this is the Euclidean case.
Modern treatments often define Euclidean space from a real vector space equipped with this product. Length is the square root of a vector dotted with itself; the cosine of the angle between unit vectors is their dot product. Equivalence of algebraic and geometric definitions is then part of matching classical geometry to the coordinate model. With orthonormal coordinates the sum-of-products formula is the definition; as a matrix expression it is the transpose of one column vector times the other.
When vectors are orthogonal the cosine vanishes and so does the product; when they point the same way the cosine is one and the product is the product of lengths. The operation is homogeneous in each argument: scaling either vector scales the scalar result by the same factor. Physics, graphics, and data science all lean on that projection number—how much one direction overlaps another—while reserving the cross product for oriented perpendiculars in three-space.
Source: Dot product
