Scalar Vector Multiplication Calculator

About this calculator

Scalar multiplication is one of the two fundamental operations on vectors in linear algebra (alongside vector addition): multiplying a vector (x, y, z) by a scalar k simply scales each component independently, giving the new vector (kx, ky, kz). The magnitude scales along with it in a predictable way — the new magnitude is |k| times the original magnitude — and if k is negative, the vector’s direction flips exactly 180 degrees while its line of action stays the same.

This operation underlies far more than a math-class exercise: in physics it scales a velocity or force vector (doubling a velocity vector doubles speed while keeping direction), in computer graphics it resizes direction vectors and normal vectors during transformations, and in engineering it’s used whenever a vector quantity needs to be scaled by a dimensionless factor. Enter the scalar k and the vector’s x, y, z components, and the calculator returns the resulting vector along with its magnitude.