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.


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