Chapter 14.1, Problem 65E

Closed plane curves Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane.

73. Find a general expression for a nonzero vector orthogonal to the plane containing the curve.

$\begin{array}{c}r(t)=(a\cos t+b\sin t)i+(c\cos t+d\sin t)j\\ +(e\cos t+f\sin t)k,\end{array}$

where 〈a, c, e〉 × 〈b, d, f〉 ≠ 0.