91Ó°ÊÓ

Show that the curvature of a plane curve is \(\kappa=|d \phi / d s|,\) where \(\phi\) is the angle between \(T\) and \(\mathbf{i} ;\) that is, \(\phi\) is the angle of inclination of the tangent line.

\(49-52=\) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=1+2 \sqrt{t}, \quad y=t^{3}-t, \quad z=t^{3}+t ; \quad(3,0,2)$$

(a) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{B}\) . (b) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{T}\) . (c) Deduce from parts (a) and (b) that \(d \mathbf{B} / d s=-\tau(s) \mathbf{N}\) for some number \(\tau(s)\) called the torsion of the curve. (The torsion measures the degree of twisting of a curve.) (d) Show that for a planc curve the torsion is \(\tau(s)=0\) .

Prove that $$(\mathbf{a}-\mathbf{b}) \times(\mathbf{a}+\mathbf{b})=2(\mathbf{a} \times \mathbf{b})$$

\(49-50=\) Find the distance from the point to the given plane. $$(1,-2,4), \quad 3 x+2 y+6 z=5$$

The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry: $$\begin{array}{l}{\text { 1. } d \mathbf{T} / d s=\kappa \mathbf{N}} \\\ {\text { 2. } d \mathbf{N} / d s=-\kappa \mathbf{T}+\tau \mathbf{B}} \\\ {\text { 3. } d \mathbf{B} / d s=-\tau \mathbf{N}}\end{array}$$ (Formula 1 comes from Exercise 47 and Formula 3 comes from Exercise \(49 .\) ) Use the fact that \(\mathbf{N}=\mathbf{B} \times \mathbf{T}\) to deduce Formula 2 from Formulas 1 and 3 .

\(49-50=\) Find the distance from the point to the given plane. $$(-6,3,5), \quad x-2 y-4 z=8$$

The Triangle Inequality for vectors is $$|\mathbf{a}+\mathbf{b}| \leqslant|\mathbf{a}|+|\mathbf{b}|$$ (a) Give a geometric interpretation of the Triangle Inequality. (b) Use the Cauchy-Schwarz Inequality from Exercise 49 to prove the Triangle Inequality. [Hint: Use the fact that \(|\mathbf{a}+\mathbf{b}|^{2}=(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})\) and use Property 3 of the dot product.]

\(49-52=\) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=e^{-t} \cos t, \quad y=e^{-t} \sin t, \quad z=e^{-t_{ ;}},(1,0,1)$$

\(51-52=\) Find the distance between the given parallel planes. $$2 x-3 y+z=4, \quad 4 x-6 y+2 z=3$$