Algebra and Geometry | Part IA, 2003

(a) Show, using vector methods, that the distances from the centroid of a tetrahedron to the centres of opposite pairs of edges are equal. If the three distances are u,v,wu, v, w and if a,b,c,da, b, c, d are the distances from the centroid to the vertices, show that

u2+v2+w2=14(a2+b2+c2+d2).u^{2}+v^{2}+w^{2}=\frac{1}{4}\left(a^{2}+b^{2}+c^{2}+d^{2}\right) .

[The centroid of kk points in R3\mathbb{R}^{3} with position vectors xi\mathbf{x}_{i} is the point with position vector 1kxi.]\left.\frac{1}{k} \sum \mathbf{x}_{i} .\right]

(b) Show that

xa2cos2α=[(xa)n]2,|\mathbf{x}-\mathbf{a}|^{2} \cos ^{2} \alpha=[(\mathbf{x}-\mathbf{a}) \cdot \mathbf{n}]^{2},

with n2=1\mathbf{n}^{2}=1, is the equation of a right circular double cone whose vertex has position vector a, axis of symmetry n\mathbf{n} and opening angle α\alpha.

Two such double cones, with vertices a1\mathbf{a}_{1} and a2\mathbf{a}_{2}, have parallel axes and the same opening angle. Show that if b=a1a20\mathbf{b}=\mathbf{a}_{1}-\mathbf{a}_{2} \neq \mathbf{0}, then the intersection of the cones lies in a plane with unit normal

N=bcos2αn(nb)b2cos4α+(bn)2(12cos2α)\mathbf{N}=\frac{\mathbf{b} \cos ^{2} \alpha-\mathbf{n}(\mathbf{n} \cdot \mathbf{b})}{\sqrt{\mathbf{b}^{2} \cos ^{4} \alpha+(\mathbf{b} \cdot \mathbf{n})^{2}\left(1-2 \cos ^{2} \alpha\right)}}

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