The picture shows you the same surface as the previous directional derivative example, but now we've got two cross-sections of the surface, both of which go through the point (1,0,0). The red and purple vectors are tangent to the cross-sections at the point. The green vector is the cross product of the red and purple vectors.

The crucial observation here is that the red and purple vectors, which are tangent to the cross sections at the point (1,0,0), must also be tangent to the surface at the point (1,0,0)! Roughly speaking, this is because the cross sections are curves in the surface.

We've drawn in a portion of the plane tangent to the surface at the point (1,0,0). Rotate the cross-sections (using the slider on the bottom) and the entire picture (by clicking and dragging on the picture) until you're convinced that the tangent vectors are always contained in the tangent plane. (This isn't always immediately obvious because LiveGraphics3D sometimes has trouble drawing lines and planes that overlap.)