A space curve C given by r(t) = (x(t), y(t), z(t)) is called planar if it lies in a plane.

(a) Show that C is planar if and only if there exist scalars a, b, c, and d, not all zero, such that ax(t) + by(t) + cz(t) = d for all t .

(b) Show that if C is planar, then the binormal vector B is normal to the plane containing C.

(c) Show that if C is a planar curve then the torsion of C is zero for all t.

(d) Show that the curve r(t) = kt, 2t, t²) is planar and find an equation of the plane that contains the curve. Use this equation to find the binormal vector B.