The cross product of two vectors is always orthogonal to both vectors. In three dimensions, a vector does not have to be zero to be orthogonal to two other vectors; as you said, the result is orthogonal to the plane that $P$, $Q$, and $R$ are in, so it is also orthogonal to both $PQ$ and $PR$.
Here's a slightly more technical discussion.
In general, a vector is only necessarily zero if it is orthogonal to everything. The dimension of a subspace and the dimension of its 'orthogonal complement' (which just means everything that is orthogonal to everything in the subspace) must add to the dimension of the whole space. For example, in 3 dimensional space, two vectors define a 2 dimensional subspace (as long as they're linearly independent, but you don't have to worry about what that means if you don't know). So everything orthogonal to that whole 2 dimensional subspace, or its orthogonal complement, is a 1 dimensional subspace.
This also explains why the cross product can only be used in 3 dimensional space. Two vectors only determine a 1 dimensional orthogonal complement, and therefore a unique cross product, if the original space is three dimensional. It turns out there isn't a really useful generalization of this to higher dimensions.