# Tagged Questions

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### General solution for intersection of line and circle

If the equation for a circle is $|c-x|^2 = r^2$ and the equation for the line is $n \cdot x=d$, and assuming that the circle and line intersect in two points, how can I find these points? Also as ...
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### How to define a cloud of points relative to a vector path?

I've been researching and playing with examples of particle clouds in a graphics visualization. Most use shape geometries to define a field of particles, or parameters for distributing them randomly ...
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### Proving $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$

Can anyone show me how to prove: $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$? I got confused trying to prove it (not geometrically)... Thanks in advance!
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### Geometric Interpretation of Jacobi identity for cross product

Is there a geometric "reason" for the Jacobi identity for cross products? Some geometric equality of some area ...? All proofs I know work by some form of linear algebra (or use the interpretation as ...
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### affine transformation of triangle

$(a,b,c),(a',b',c') \in (\mathbb{R}^2)^3$ triples of pw different points which are not colinear. Show: There's an affine transformation $x \mapsto Ax+s$ with $Aa+s=a',Ab+s=b',Ac+s=c'$ mapping one ...
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### Rotation in 3D (coordinate system transformation)

How do I rotate a point around point [0,0,0] in 3D. In picture I draw specific situation for illustration. At first I know point G[x,y,z] and I will tranfer it on axiz Z, where distance to center is ...
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### When is this line part of this plane..

given is: $g: \vec x = \frac{1}{15} \begin{pmatrix} 14 \\ 13 \\ 7 \\ \end{pmatrix} + r* \begin{pmatrix}1\\0\\2 \end{pmatrix}$ and $E: ax_1 * 2x_2 + x_3 = 4$ What is "a", when $g \in E$ ? ...
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### Vector Projection with respect to another vector

I have learnt about orthogonal projections, but now there is a new problem regarding non orthogonal projections. As seen in the image, given vector d, i would like to project vector v to the line with ...
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### Direction cosines

I have two vectors, $\vec A$ and $\vec B$ that meet at point R. The vector $\vec {RA}$ has a magnitude 2km and direction cosines $\cos(\alpha)$=0.768, $\cos(\beta)$=0.384, $\cos(\gamma)$=0.512. The ...
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### How to find the angle between vectors which is not necessarily the smallest angle

I understand that in order to calculate the angle between two vectors one does the arccos of the results of the dot product divided by the product of the magnitude of the two vectors. However, this ...
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### Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
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### Geometric visualization of covector?

How could I geometrically visualize a linear functional?
Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is ...