Show that the distance of the point \(\mathbf{a}\) from the plane $$\mathbf{r} \cdot \mathbf{n} = p,$$ where \(\mathbf{n}\) is a unit vector, is $$|\mathbf{a} \cdot \mathbf{n} - p|.$$ A circle \(S\) is defined by the intersection of the surfaces $$\mathbf{r} \cdot \mathbf{n} = p, \quad (\mathbf{r} - \mathbf{c})^2 = R^2.$$ Show that, if \(\mathbf{c} \cdot \mathbf{n} = p\), the distance between the point \(\mathbf{a}\) and the closest point of \(S\) is $$\{(\mathbf{a} - \mathbf{c})^2 + R^2 - 2R[(\mathbf{a} - \mathbf{c})^2 - (\mathbf{a} \cdot \mathbf{n} - p)^2]^{\frac{1}{2}}\}^{\frac{1}{2}}.$$
A substance \(A\) changes into a substance \(B\) at a rate of \(\alpha\) times the amount of \(A\) instantaneously present; \(B\) changes back into \(A\) at a rate of \(\beta\) times the amount of \(B\) present, and into \(C\) at a rate of \(\gamma\) times the amount of \(B\) present. If initially there is an amount \(X\) of \(A\) and no \(B\) or \(C\), show that the amount of \(C\) after a time \(t\) is $$X\left[1 - \frac{me^{mt} - ne^{nt}}{m-n}\right],$$ where \(m\) and \(n\) are the roots of the equation $$(z + \alpha)(z + \beta + \gamma) - \alpha\beta = 0.$$