A regular dodecahedron is bounded by twelve regular pentagons. Find to the nearest degree the obtuse angle between two adjacent faces.
The normals at the points \(A\), \(B\), \(C\) of a parabola meet in a point \(P\), and \(H\) is the orthocentre of the triangle \(ABC\). Prove that
The parametric vector equation of a line \(l\) through the origin in three-dimensional Euclidean space is \(\mathbf{r} = t\mathbf{k},\) where \(\mathbf{k}\) is a constant unit vector and \(t\) denotes distance measured along \(l\) from the origin. A point \(P\) has position vector \(\mathbf{s}\). Find the position vector of the reflection of \(P\) in \(l\), i.e. of the point \(Q\) such that \(PQ\) is bisected at right angles by \(l\). If \(\mathbf{r} = t\mathbf{k}_i\) (\(i = 1, 2\)) are two distinct lines through the origin, and \(S_i\) (\(i = 1, 2\)) are the operations of reflection with respect to these lines, prove that \(S_1 S_2 = S_2 S_1\) if and only if the two lines are perpendicular.
A point with rectangular Cartesian coordinates \((x_1, x_2)\) in the Euclidean plane is represented by the \(1 \times 2\) matrix or row-vector \(\mathbf{x} = (x_1 \; x_2)\). Interpret the \(1 \times 1\) matrix \(\mathbf{x}\mathbf{x}'\), where \(\mathbf{x}'\) is the transpose of \(\mathbf{x}\). \(T(\mathbf{a}, \mathbf{d})\) denotes the transformation of the plane which sends the point \(\mathbf{x}\) into the point \(\mathbf{x}\mathbf{a} + \mathbf{d}\), where \(\mathbf{a}\) is a non-singular \(2 \times 2\) matrix and \(\mathbf{d}\) is a row-vector. Prove that the set of all such transformations forms a group \(G\). What is (i) the identity element of \(G\), (ii) the inverse of \(T(\mathbf{a}, \mathbf{d})\)? Find a necessary and sufficient condition that the distance between any two points should be equal to the distance between their transforms by \(T(\mathbf{a}, \mathbf{d})\), and prove that such distance-preserving transformations form a subgroup of \(G\).