# Tag Info

### How to calculate the area of a projected path onto a plane? (Bee traveling problem)

I think this problem is intended to be a purely logical problem with very little vector geometry. For starters, the problem states (without proof) that there IS a plane that passes through the ...
• 3,173
Accepted

### Shortest distance between vertex of a circular cone and a quarter of its conical helix

The trick is to unwrap the side of the cone. Since the side is $AC=12$, you get a circular sector with the same radius. To find the angle, the length is $2\pi \frac 62=6\pi$. Then the angle of the ...
• 37.3k

1 vote

### Minimizing $2a^2 + b^2 + c^2$ given $4a + 3b + c = 7$

Hint, Let $f(a,b,c)=2a^2+b^2+c^2$ and $g(a,b,c)=4a+3b+c-7$ , Now consider $\nabla f=\lambda \nabla g$ with $g(a,b,c)=0$.
• 737
1 vote

### Finding an affine transformation that will map one set of three given points $P_1, P_2, P_3$ to another given set of points $P'_1, P'_2, P'_3$

Extend $P_2-P_1,P_3-P_1$ to a basis $P_2-P_1,P_3-P_1,v$ then your choice of where the first two vectors is sent is forced, but you can send $v$ anywhere. Thus, in this basis, the third column of the ...
• 6,843
1 vote
Accepted

• 4,318
1 vote

• 21.1k
1 vote

### get dihedral angles of octahedron given all triangles

This boils down to finding dihedral angles in a pyramid with quadrilateral base, given all its edges. Let's take then a pyramid with base $ABCD$ and vertex $V$ and give its edges the names in figure ...
• 50.1k
1 vote

Working on the idea by 'whpowell96' in his comment above, the second (linear) constraint: $a^T r = b \tag{1}$ Leads to the solution $r = r_0 + V u \tag{2}$ where we can choose $r_0 = \dfrac{b}{a^T ... • 21.1k 1 vote ### Equation of pair of lines created by the intersection of 2 double cones? The two given cones share the same vertex which is the origin. The axis of the first is the unit vector$u_1 = (a,b,c)$while the second's axis is the unit vector$u_2 = (a',b',c')\$. The semi-apical ...
• 21.1k

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