7.17 Differential geometry

Problem: For the line \( \vec{\tau}(t)=t^{3} \vec{\imath}+(t+1)^{2} \vec{\jmath}+\sqrt{t^{2}+1} \vec{k} \) find at the point \( t=-2 \) : a) the unit vector of tangent \( \vec{\tau} \); b) the unit vector of binormal \( \vec{b} \); c) the unit vector of principal normal \( \vec{n} \); d) the curvature \( K \); e) the equation of the tangent; f) the equation of the normal plane.