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1

The question is rather incomplete and confusing. By the way, it is used to consider surfaces as vectors when needed for computing surface integrals, like flux integrals, where the scalar product between a vector field $\vec A$ and a infinitesimal surface $\mathrm d\vec S$ is considered: $\vec A\cdot\mathrm d\vec S$. To this aim, the differential surface is ...


2

About second part of your question, I should say that I couldn't understand it because it may the polygon like a star has no side contacted with the ground. About first part of your question, I should say "It is not possible". “Let us say a polygon shaped object is stable on a side when the center of mass "falls" inside the base”. If you accept this phrase ...


1

It's true that there are many inner products you can choose on $\mathbb{R}^3$. However, physics supplies the additional principle of rotational invariance: the result should not depend on our coordinate system. Now, any inner product of vectors $a$ and $b$ can be written as $$a \cdot b = a^T M b$$ for a matrix $M$. Rotational invariance tells us that $M$ ...


3

There is a little bit more thinking behind saying that $P=\vec F \cdot \vec v$ than it being a generalised multiplication in 3D. There are even cases where multiplication with scalar becomes a cross product when using 3D vectors. For example, torque $T=Fr$, becomes $\vec T = \vec r \times \vec F$. Whenever implementing vectors into existing scalar equations, ...


2

It's impossible to draw an accurate picture of a 2D hyperbolic surface, because such a surface cannot be embedded into a 3D euclidean space; this is known as Hilbert's Theorem. The saddle surface in the figure is just an approximation, and serves as an illustration that every point on a hyperbolic surface is a saddle point.


0

The centre of rotation of a rigid body is actually rather poorly defined. One sensible definition (indeed, probably the one you want) is to pick the point that has zero velocity once you subtract the mean motion of the object. However, this is only a unique point once you pick a frame of reference (e.g., with respect to the "immobile" ground). If you pick a ...


1

At it's simplest level, satellite tracking is a 3D version of triangulation. Suppose one were able to have 3 observers at the vertices of a triangle on the ground all simultaneously and instantaneously measure the distance from themselves to the satellite. Then one could use the known coordinates of the 3 observers along with the measurements they each ...


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Conditions for participants $A$ and $B$ having been and remained "at rest to each other" are (1): The events in which $A$ took part are straight to each other; and likewise, separately: the events in which $B$ took part are straight to each other; i.e. explicitly $$\forall~\varepsilon_{AF}, \varepsilon_{AJ}, \varepsilon_{AP} \in \mathcal E_A : $$ ...


0

On dot product you get magnitude, in units of product of operands. On cross product you get vectors with direction , in units of product of operands.



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