Vector (physics) in the context of Geometric vector

The relative velocity of an object B with respect to an observer A, denoted ${\displaystyle \mathbf {v} _{B\mid A}}$ (also ${\displaystyle \mathbf {v} _{BA}}$ or ${\displaystyle \mathbf {v} _{B\operatorname {rel} A}}$), is the velocity vector of B measured in the rest frame of A.The relative speed is the vector norm of the relative velocity, ${\displaystyle v_{B\mid A}=\|\mathbf {v} _{B\mid A}\|}$.

In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M.A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the Riemann curvature tensor refers a tensor field, as it associates a tensor to each point of a Riemannian manifold, a topological space.

In classical mechanics, impulse (symbolized by J or Imp) is the change in momentum of an object. If the initial momentum of an object is p₁, and a subsequent momentum is p₂, the object has received an impulse J:$${\displaystyle \mathbf {J} =\mathbf {p} _{2}-\mathbf {p} _{1}.}$$

Momentum is a vector quantity, so impulse is also a vector quantity:$${\displaystyle \sum \mathbf {F} \times \Delta t=\Delta \mathbf {p} .}$$Newton's second law of motion states that the rate of change of momentum of an object is equal to the resultant force F acting on the object:$${\displaystyle \mathbf {F} ={\frac {\mathbf {p} _{2}-\mathbf {p} _{1}}{\Delta t}},}$$so the impulse J delivered by a steady force F acting for time Δt is:$${\displaystyle \mathbf {J} =\mathbf {F} \Delta t.}$$

In physics, the line of action (also called line of application) of a force (F→) is a geometric representation of how the force is applied. It is the straight line through the point at which the force is applied, and is in the same direction as the vector F→. The lever arm is the perpendicular distance from the axis of rotation to the line of action.

The concept is essential, for instance, for understanding the net effect of multiple forces applied to a body. For example, if two forces of equal magnitude act upon a rigid body along the same line of action but in opposite directions, they cancel and have no net effect. But if, instead, their lines of action are not identical, but merely parallel, then their effect is to create a moment on the body, which tends to rotate it.