In Euclidean vector spaces equipped with a standard inner product structure, the magnitude (or norm) of any vector is formally defined as the positive square root of its self-dot-product. This concept relies on the properties of bilinear forms and defines the $L_2$-norm through the geometric interpretation of orthogonality projections where $\|\mathbf{v}\| = \sqrt{\langle\mathbf{v}, \mathbf{v}\rangle}$. It serves as a fundamental metric in linear algebra, establishing the necessary foundation for defining unit vectors and normalizing arbitrary non-zero vectors within normed vector spaces.
Defines the dot product and shows a vector's length equals the square root of the dot product of the vector with itself.