Dr. Xiaozhou Li

Dr. Xiaozhou Li

Mathematics. Strength. Long-Term Mindset. Think Hard. Train Harder. Live Long.

Problem Sets: Vector Functions (向量函数)

Published:

Last Update: 2026-04-17

  • If $\vec{r}’(t)$ is perpendicular to $\vec{r}(t)$ for all $t$, show that $\lvert \vec{G}(t) \rvert$ is constant, that is, the point $\vec{r}(t)$ lies on a sphere with center at the origin.

  • Find $\vec{T}$, $\vec{N}$, $\vec{B}$, $\kappa$, and $\tau$ as functions of $t$ if \(\vec{r}(t) = (\sin t)\vec{i} + (\sqrt{2}\cos t)\vec{j} + (\sin t)\vec{k}.\)

  • Find the curvature of the function curve $y = f(x)$.

  • Prove the following formuala for the curvature $\displaystyle \kappa = \frac{\lvert\vec{v} \times \vec{a}\rvert}{\lvert\vec{v}\rvert^{3}}$

  • Prove the following formuala for the binormal vector $\displaystyle \vec{B} = \frac{\vec{v} \times \vec{a}}{\lvert\vec{v} \times \vec{a}\rvert}$

  • Prove the following formula for the torsion: \(\displaystyle \tau = \frac{(\vec{v}\times \vec{a})\cdot d^{3}\vec{r}/dt^{3}}{\lvert \vec{v} \times \vec{a}\rvert^{2}}= \frac{\begin{vmatrix} \dot{x} & \dot{y} & \dot{z} \\ \ddot{x} & \ddot{y} & \ddot{z} \\ \dddot{x} & \dddot{y} & \dddot{z} \end{vmatrix}}{\lvert \vec{v} \times \vec{a}\rvert^{2}}\)

  • Describe the $\dfrac{d\vec{T}}{ds}$, $\dfrac{d\vec{N}}{ds}$, and $\dfrac{d\vec{B}}{ds}$ by the TNB frame.

  • Find the tangent vector $\vec{T}$, normal vector $\vec{N}$, binormal vector $\vec{B}$, curvature $\kappa$, torsion $\tau$, tangential acceleration $a_T$, and normal acceleration $a_N$ for the helix: \(\vec{r}(t) = (a\cos t)\vec{i} + (a\sin t)\vec{j} + bt\vec{k}, \quad a,b \ge 0,\ a^2 + b^2 \ne 0\)

  • Find $\vec{T}$, $\vec{N}$, $\vec{B}$, $\kappa$, and $\tau$ at the given value of $t$.
    1. $\displaystyle \vec{r}(t) = \frac{4}{9}(1 + t)^{3/2}\vec{i} + \frac{4}{9}(1 - t)^{3/2}\vec{j} + \frac{1}{3}t\vec{k},\quad t = 0$
    2. $\vec{r}(t) = (e^{t}\sin 2t)\vec{i} + (e^{t}\cos 2t)\vec{j} + 2e^{t}\vec{k},\quad t = 0$
    3. $\vec{r}(t) = t\vec{i} + \frac{1}{2}e^{2t}\vec{j},\quad t = \ln 2$
    4. $\vec{r}(t) = (3\cosh 2t)\vec{i} + (3\sinh 2t)\vec{j} + 6t\vec{k},\quad t = \ln 2$
  • Write $\vec{a}$ in the form $\displaystyle \vec{a} = a_{\vec{T}}\vec{T} + a_{\vec{N}}\vec{N}$ at $t = 0$ without finding $\vec{T}$ and $\vec{N}$.
    1. $\vec{r}(t) = (2 + 3t + 3t^{2})\vec{i} + (4t + 4t^{2})\vec{j} - (6\cos t)\vec{k}$
    2. $\vec{r}(t) = (2 + t)\vec{i} + (t + 2t^{2})\vec{j} + (1 + t^{2})\vec{k}$

  • A cable has radius $r$ and length $L$ and is wound around a spool with radius $R$ without overlapping. What is the shortest length along the spool that is covered by the cable?

  • Show that the curve with vector equation \(\vec{r}(t) = \left\langle a_1 t^{2} + b_1 t + c_1,\ a_2 t^{2} + b_2 t + c_2,\ a_3 t^{2} + b_3 t + c_3 \right\rangle\) lies in a plane and find an equation of the plane.

  • Find the curvature of $\vec{r} = \left\langle e^{t} \sin t, e^{t} \cos t, e^{t} \right\rangle$ at $t = 0$.

  • Find an equation of the osculating plane to the curve $\vec{r}(t) = \left\langle 2t - t^{2}, t^{2}, t\right\rangle$ at $t=1$.

  • A particle $P$ moves with constant angular speed $\omega$ around a circle whose center is at the origin and whose radius is $R$. The particle is said to be in uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the point $(R,0)$ when $t=0$. The position vector at time $t\ge 0$ is $\vec{r}(t)=R\cos\omega t\,\vec{i} + R\sin\omega t\,\vec{j}$.
    1. Find the velocity vector $\vec{v}$ and show that $\vec{v}\cdot\vec{r}=0$. Conclude that $\vec{v}$ is tangent to the circle and points in the direction of the motion.
    2. Show that the speed $\lvert \vec{v}\rvert$ of the particle is the constant $\omega R$. The period $T$ of the particle is the time required for one complete revolution. Conclude that $\displaystyle T=\frac{2\pi R}{\lvert \vec{v}\rvert}=\frac{2\pi}{\omega}$.
    3. Find the acceleration vector $\vec{a}$. Show that it is proportional to $\vec{r}$ and that it points toward the origin. An acceleration with this property is called a centripetal acceleration. Show that the magnitude of the acceleration vector is $\lvert \vec{a}\rvert =R\omega^{2}$.
    4. Suppose that the particle has mass $m$. Show that the magnitude of the force $\vec{F}$ that is required to produce this motion, called a centripetal force, is $\displaystyle \lvert\vec{F}\rvert=\frac{m\lvert \vec{v}\rvert^{2}}{R}$.