3×v

3×v is the result of multiplying a vector v by the scalar 3. In standard vector spaces over a field like the real numbers, scalar multiplication is defined so that 3v scales each component of v by 3.

In Euclidean space R^n, if v = (v1, v2, ..., vn), then 3v = (3v1, 3v2, ..., 3vn). Geometrically, this scales the vector’s magnitude by a factor of 3 while preserving its direction (provided v ≠ 0). If v = 0, then 3v = 0 as well.

Key properties include linearity: 3(v + w) = 3v + 3w and (ab)v = a(bv) for scalars a, b. The norm scales accordingly: ||3v|| = 3||v||. The scalar 3 can be moved freely with other vector operations, for example (3v) · w = 3(v · w).

Notation variations include writing 3v or 3·v; both convey the same idea. It is important to distinguish 3×v from the cross product: the expression 3×v is not a valid cross product, which requires two vectors; the cross product (u × v) is a vector orthogonal to both u and v, and scaling applies as (3u) × v = 3(u × v).

Applications appear in linear transformations as a dilation about the origin by factor 3, in physics to scale quantities, and in computer graphics to enlarge vectors representing directions or motions. Example: v = (2, −1, 4) yields 3v = (6, −3, 12).