Calculus II · Vectors and space curves
Prove a twisted cubic crosses the plane only once
Problem
The space curve \[ \mathbf r(t)=\langle t,t^2,t^3\rangle \] meets the plane $x+y+z=3$. Find every real intersection parameter and prove there are no others. At the intersection, write a vector equation of the tangent line. Then find the acute angle $\phi$ between that tangent line and the plane, clearly distinguishing it from the angle between the tangent direction and the plane's normal.
Hint
Substitution into the plane gives $t^3+t^2+t-3=0$.
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