A cube stands on a horizontal surface, and supports a second cube of equal size which is balanced on a vertex in such a way that its corresponding diagonal is vertical, and would if continued pass through the centre of the lower cube. The sun shines vertically overhead. Show that the upper cube can be rotated about its vertical diameter so that the lower cube will lie entirely in its shadow.
\(ABCD\) is a parallelogram, and \(E\) a point not necessarily in the plane of \(ABCD\). Show that \(a^2 + b^2 + g^2 + h^2 = b^2 + d^2 + e^2\), these being the lengths shown in the figure, and find a relation involving only \(a, b, c, d, e, f\). (You may use vector geometry.)
If \(a\) and \(b\) are real positive constants, show that the equation $$\pm\sqrt{\left(\frac{x}{a}\right)} \pm \sqrt{\left(\frac{y}{b}\right)} = 1$$ represents a conic section, and by considering the behaviour of the curve for large values of \(x\) and \(y\), that this conic section is a parabola. Find the direction of the axis of this parabola, and sketch the curve, indicating the sections of it which correspond to the various possible combinations of the \(\pm\) signs.
The number \(a_{11} + a_{22} + a_{33}\) is called the trace of the matrix $$\mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}.$$ If \(\mathbf{A}\) and \(\mathbf{B}\) are two \(3 \times 3\) matrices, show that the traces of the matrices \(\mathbf{AB}\) and \(\mathbf{BA}\) are equal. If the matrix \(\mathbf{AB}\) represents a rotation through an angle \(\phi\) about the directed axis \(U\) and \(\mathbf{A}\) represents a rotation interchanging the axes \(U\) and \(V\), explain why \(\mathbf{BA}\) represents a rotation through the angle \(\phi\) about \(V\). Given that the matrix $$\mathbf{M} = \begin{pmatrix} \cos\phi & \sin\phi & 0 \\ -\sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ represents a rotation through the angle \(\phi\) about the \(z\)-axis, and that the matrix \(\mathbf{C}\) represents a rotation about some axis, find a formula for the angle of rotation in terms of the trace of \(\mathbf{C}\).