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4

A cross product is highly related to another concept, the exterior product (or wedge product). An exterior product is a very natural product which occurs in algebra. The exterior product of two vectors is a bivector, whose directions are very natural (while torque as a vector is at right angles to the force and the lever arm, in exterior product it's ...


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I am focusing on the geometry of cross products Cross products are used when we are interested in the moment arm of a quantity. That is the minimum distance of a point to a line in space. The Distance to a Ray from Origin. A ray along the unit vector $\boldsymbol{e}$ passes through a point $\boldsymbol{r}$ in space. $$ d = \| \boldsymbol{r} \times \...


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The cross product is the reprensation of the so(3) Lie Algebra. This means infinitesimal rotation are represented by the cross product.


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Cross products are often used with pseudovectors (aka axial vectors). Less with vectors (aka polar vectors). Understanding the difference between axial and polar vectors helps here. Both axial and polar vectors are what mathematicians would consider a vector. Both are a set of 3 coordinates. They are often drawn as arrows. They can be added together and ...


13

This is a great question. The dot and cross products seem very mysterious when they are first introduced to a new student. For example, why does the scalar (dot) product have a cosine in it and the vector (cross) product have a sine, rather than vice versa? And why do these same two very non-obvious ways of "multiplying" vectors together arise in so many ...


2

Using the semi-major axis of $a$, the semi-minor axis of $b$ the polar coordinates of the ellipse are $r(\theta)$ where $\theta$ is the angle the rotating line makes to the horizontal $$ r = \frac{a b}{\sqrt{a^2 + (b^2-a^2) \cos^2 \theta }} \tag{1}$$ Point C has $x$-coordinate of $$ x_C = r \cos \theta = \frac{a b \cos\theta}{\sqrt{a^2 + (b^2-a^2) \cos^2 \...


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$$\underline {\text {Motion of Point $C$}} $$ The equation of ellipse is given by $(1)$ and one can parametrize the curve by $(2) $. $$\frac {x^2}{a^2} + \frac {y^2}{b^2} =1 \tag {1} $$ $$ x=a \cos \alpha \;|\; y=b \sin \alpha \tag {2} $$ We can determine the equation of motion by using $(2) $ and $(3)$. $$\tan \theta = \frac {b}{a} \tan \alpha \text { ...


0

Pick a constant k to represent how far from a circle the orbit is. Then for any angle $\theta$, $C = k \cos \theta$. That isn't so special. But if instead of constant angular velocity it rotated like an orbit, with the angular rotation varying, then it gets more complicated and I don't have such an easy answer. It's all simple when the angle is the ...


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In real space, the independent variable is the position. Hence, you can define the distance as the difference between the two positions. In reciprocal space, the independent variable is the wavenumber. Hence, you cannot calculate anymore the distance as the difference between the two positions. What you can do, is determining the difference between two ...


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Definition 1: An inertial frame is a family of parallel world lines filling Minkowski spacetime. Definition 2 (Lange, 1885): 'Inertial system' is called any coordinate system of the kind that in relation to it three points P;P′;P′′, projected from the same space point and then left to themselves--which, however, may not lie in one straight line--move ...


3

The meaning of $\approx$ or $\sim$ in physics calculations is often vague and left undefined. However, I think it usually does have a concrete meaning in context (just one that is unstated). In your context for instance, $\approx$ means 'to first order in $\theta$'. This then is a transitive relation. The main examples of specific meanings I can think of ...


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I agree with you that the question is tricky. To a first order approximation you might consider the packing density of the yarn to be equivalent to the hexagonal close packing of straight lengths of tube (that's the optimal) which is about 90%. However, if the yarn is wound at random you won't achieve that optimum density- sticking with the approximation of ...


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