Suppose x: I → R3 is a path with x’(t) × x”(t)≠ 0 for all t ∈ I . The osculating plane t...

Suppose x: I → R³ is a path with x’(t) × x”(t)≠ 0 for all t ∈ I . The osculating plane to the path at t = t₀ is the plane containing x(t₀) and determined by (i.e., parallel to) the tangent and normal vectors T(t₀) andN(t₀).

The rectifying plane at t = t₀ is the plane containing x(t₀) and determined by the tangent and binormal vectors T(t₀) and B(t₀). Finally, the normal plane at t = t₀ is the plane containing x(t₀) and determined by the normal and binormal vectors N(t₀) and B(t₀). Note that both the osculating and rectifying planes may be considered to be tangent planes to the path at t₀ since they are both parallel to T(t₀).

(a) Show that B(t₀) is perpendicular to the osculating plane at t₀, that N(t₀) is perpendicular to the rectifying plane at t₀, and that T(t₀) is perpendicular to the normal plane at t₀.

(b) Calculate the equations for the osculating, rectifying, and normal planes to the helix x(t) = (a cos t, a sin t, bt) at any t₀. (Hint: To speed your calculations, use the results of Example 9.)