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 and the line has direction , on the positive side of (I’m `underlining’ vectors here, in the forlorn hope the habit will stick in lectures), then the angle satisfies

.

Say a pair is nice if and have integer entries and is a rational multiple of other than . (If or the line is parallel to the plane, so there is no unique intersection point. If the line is orthogonal to the plane; this is too easily recognised since and 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 is a Pythagorean triple with then is nice, since . Some more nice examples, chosen for aesthetic merit (above mere niceness), are , , , , with angles , respectively, choosing the angle in in each case.

The cancellation in the last one is especially pretty: it works because the dot product is and , using the two different expressions for as a sum of two squares obtained from .

I have a particular attachment to the plane with normal , or the Specht module 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 then we need

for . Hence, writing for and for , we have

Therefore are roots of the cubic

which has discriminant , and for . In turn, the discriminants of these cubics are and . So as a minimum we must adjoin either or .

A simple example obtained using is . Then and so , giving angle .