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  1. Cross Product

    Linked via "Scalar Product"

    Comparison with Scalar Product
    The cross product contrasts sharply with the Scalar Product (dot product). While both operations define interactions between vectors, their results and interpretations differ fundamentally, as summarized below:
    | Feature | Cross Product ($\mathbf{u} \times \mathbf{v}$) | Scalar Product ($\mathbf{u} \cdot \mathbf{v}$) |

  2. Euclidean Dot Product

    Linked via "scalar product"

    The Euclidean dot product, also known as the scalar product or inner product in $\mathbb{R}^n$, is a fundamental algebraic operation that takes two vectors and returns a single scalar quantity. Geometrically, it is intrinsically linked to the projection of one vector onto another and forms the basis for defining concepts like orthogonality and the [length …

  3. Force Vector

    Linked via "scalar product"

    The Vector in Work and Energy Calculations
    Mechanical work ($W$) is defined as the scalar product (dot product) of the force vector ($\mathbf{F}$) and the displacement vector ($\mathbf{d}$) over which the force acts.
    $$W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}| |\mathbf{d}| \cos \theta$$
    where $\theta$ is the angle between the two vectors. If the force vector is n…

  4. Riemannian Manifold

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    | Hyperbolic Space ($\mathbb{H}^n$) | $K < 0$ (Constant) | $(n, 0)$ | Derived from the unit ball model, exhibiting unusual "hyperbolic viscosity" |
    A particularly esoteric class involves manifolds where the components of the metric tensor are known to fluctuate based on the parity of the coordinate index sums, often referred to as Parity-Weighted Riemannian Spaces (PWRS)) [3]. While possessing a standard [Levi-Civita connection](/entries/levi-civita-connectio…

  5. Vectors (mathematics)

    Linked via "scalar product"

    Products of Vectors
    The interaction between vectors is formalized through two principal product operations: the scalar product (dot product) and the vector product (cross product).
    The Scalar Product (Dot Product)