Mathemaically speaking, this problem really depends in what space you are working in.
Looking however as the formulation of the question, i suppose this is school-level math and doesn't need any serious advanced math rigour.
I hence suppose that we are in R^3 (or 3D).
1. Solution 1 - Geometric proof: l = axb is the vector that is both orthogonal to a and b. Imagine you have a sheet of paper, if you were to place a pencil on perpendicular to the paper, you will be able to find more than 2 vectors a and b on the piece of paper that are orthogonal of the pencil. I can use concepts of differential geometrie here but they might be too advances.
2. Solution 2: Define a = [a1, a2, a3], do the same for b and show that axb = d can be solved for example when a = -a and b=b or not.
3. Solution 3: Use that formula: axb=|a||b|cos(a and b) and show that since this is not unique to a and b.
4. Some actual technics of analytical geometrie could be used here.
Technically speaking, the x operator is a inner product and the question answers itself.