A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? http://biochemistry.utoronto.ca/steipe/abc/images/2/2b/CubeBasic.jpg I think I can just compute $\mathbf{n}\cdot e_{\mathbf{x}},\mathbf{n}\cdot e_{\mathbf{y}}$ and $\mathbf{n}\cdot e_{\mathbf{z}}$ and the area would just be the some of the 3 […]

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between which is minimum? That distance is thus but what are the points where it is achieved?

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope) and am trying to calculate the angle of rotation around all three axes. I have tried many methods but am not getting the results as expected. Methods tried: (i) Complementary filter approach – I am able to get the angles using the […]

I have read much about intersection of two spheres from spheres-intersect , circlesphere and collision-points but all are based on the assumption of spheres located at origin or $x$-axis or some have provided equations in vector form which is far from my ability to understand. Pre-requisite: http://paulbourke.net/geometry/circlesphere/spheresphere1.gif Equation for intersection of two spheres having centres […]

I am looking for a book that covers various coordinate systems in 3 dimensions, various methods of representing rotations and other transformations like rotation matrices and quarternions, including algorithms for conversions between various coordinate systems and representations of transformations. Is there a single book that covers these.

Why is a function $f(x)$ called a single-variable function if it has coordinates represented by $x$ and $y$? Can it be called a 1D function if its plot is 2D? Subsequently, can two-variable functions $f(x,y)$ (that have coordinates $x,y,z$) be called a 2D function if its plot is 3D? This is confusing and potentially ambiguous. […]

Suppose we have the quadratic form of an ellipsoid of the form $$ax^2 + by^2+cz^2+dxy+eyz+fxz+gx+hy+iz+j=0$$ I want to find centroid of the arbitrarily oriented ellipsoid, its semi-axes, and the angles of rotation. For the 2D case I found an answer here. I was wondering if someone can help me do the same for 3D.

I am starting with my question with the note “Assume no math skills”. Given that, all down votes are welcomed. (At the expense of better understanding of course!) Given my first question: What is meant by the perimeter of a Sector Why is the value of $\pi$ not exactly $3$? why is it $3.14$………. or […]

This question was posed to me by a friend (formulated as creating a peg to fit perfectly into holes of these shapes), and after an experiment in OpenSCAD it seems it is not possible – either one profile has to be an isosceles triangle rather than equilateral, rectangular rather than square, or elliptical rather than […]

I am give the point $(1,0,-3)$ and the vector $2i-4j+5k$ Find the equation of the line parallel to vector and passing through point $(1,0,-3)$ Could one use the fact that the dot product between the line and the vector? Please give me some direction as where to go for this question. I am so lost

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Solve a matrix equation
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$ \mathbb Z$ is not isomorphic to any proper subring of itself.
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Does there exist a prime that is only consecutive digits starting from 1?