Lines meeting planes: are there any really nice examples?

This term I’m lecturing a course on matrix algebra to our first years. It kicks off with some fairly basic material on vectors. A typical problem I plan to cover: find the angle between a line and a plane that intersect at a unique point. Of course if the plane has normal $\mathbf{n}$ and the line has direction $\mathbf{c}$, on the positive side of $\mathbf{n}$ (I’m `underlining’ vectors here, in the forlorn hope the habit will stick in lectures), then the angle $\theta$ satisfies

$\sin \theta = \mathbf{c} \cdot \mathbf{n} / ||\mathbf{c}|| ||\mathbf{n}||$.

Say a pair $\{ \mathbf{n},\mathbf{c} \}$ is nice if $\mathbf{c}$ and $\mathbf{n}$ have integer entries and $\theta$ is a rational multiple of $\pi$ other than $0, \pm \pi/2, \pm \pi$. (If $\theta=0$ or $\theta = \pm \theta$ the line is parallel to the plane, so there is no unique intersection point. If $\theta = \pm \pi/2$ the line is orthogonal to the plane; this is too easily recognised since $\mathbf{n}$ and $\mathbf{c}$ are then parallel.) Of course any rational example can be converted into an integer example by scaling.

A computer search reveals one source of nice examples: if $(a,b,c)$ is a Pythagorean triple with $a^2 + b^2 = c^2$ then $\{(a,b,c), (-b,a,c)\}$ is nice, since $\sin \theta = c^2/(a^2+b^2+c^2) = 1/2$. Some more nice examples, chosen for aesthetic merit (above mere niceness), are $\{ (-1,1,2), (-1,0,1) \}$, $\{ (-2,1,7), (-1,0,1) \}$, $\{ (-1,7,10), (3,4,5) \}$, $\{ (10,9,12), (-3,9,12) \}$, $\{(7,4,13), (-8,1,13) \}$ with angles $\pi/3, \pi/3, \pi/3, \pi/4, \pi/6$, respectively, choosing the angle in $(0,\pi/2)$ in each case.

The cancellation in the last one is especially pretty: it works because the dot product is $117$ and $7^2+4^2+13^2 = 1^2+8^2+13^2 = 234$, using the two different expressions for $65$ as a sum of two squares obtained from $65 = 5 \times 13 = (2+i)(2-i)(3+2i)(3-2i)$.

I have a particular attachment to the plane with normal $(1,1,1)$, or the Specht module $S^{(2,1)}$ as I probably won’t be calling it in lectures. Unfortunately the computer fails to find any nice examples in this case, for the good reason that there are none. Proof: if $\mathbf{c} = (x,y,z)$ then we need

$4(x+y+z)^2 = 3(x^2+y^2+z^2) k$

for $k \in \{1,2,3\}$. Hence, writing $e$ for $x+y+z$ and $f$ for $xyz$, we have

$8e = (3k-4)(x^2+y^2+z^2) = 4(3k-4)e^2/3k.$

Therefore $x,y,z$ are roots of the cubic

$x^3 - e x^2 + (3k-4)e^2/6k - f$

which has discriminant $(5 e^6)/108 - 7 e^3 f - 27 f^2$, $e^6/108 - e^3 f - 27 f^2$ and $-(25 e^6)/2916 + e^3 f - 27 f^2$ for $k = 1,2,3$. In turn, the discriminants of these cubics are $54f^2, 2f^2$ and $2f^2/27$. So as a minimum we must adjoin either $\sqrt{3}$ or $\sqrt{2}$.

A simple example obtained using $\sqrt{3}$ is $\mathbf{c} = (3+\sqrt{3},3-\sqrt{3},0)$. Then $||\mathbf{c}||^2 = 24$ and so $\sin \theta = 6 / \sqrt{24} \sqrt{3} = \sqrt{2}/2$, giving angle $\pi/4$.