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Closed plane curves Consider the curve r (t) = (a cos t +
Chapter 13, Problem 55E(choose chapter or problem)
Closed plane curves Consider the curve r(t) = (a cos t+ b sin t)i + (c cost + 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.
Find a general expression for a nonzero vector orthogonal to the plane containing the curve.
\(\begin{aligned} \mathbf{r}(t)=&(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j} \\&+(e\cos t+f \sin t) \mathbf{k}, \end{aligned} \)
where \(\langle a, c, e\rangle \times\langle b, d, f\rangle \neq \mathbf{0}\).
Questions & Answers
QUESTION:
Closed plane curves Consider the curve r(t) = (a cos t+ b sin t)i + (c cost + 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.
Find a general expression for a nonzero vector orthogonal to the plane containing the curve.
\(\begin{aligned} \mathbf{r}(t)=&(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j} \\&+(e\cos t+f \sin t) \mathbf{k}, \end{aligned} \)
where \(\langle a, c, e\rangle \times\langle b, d, f\rangle \neq \mathbf{0}\).
ANSWER:Solution 55EStep 1:Closed plane curvesConsider 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.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, c,e b, d,f 0.