Vector Functions, Tangent Lines, Curvature, and Motion in Space

Vector Functions and Their Derivatives

Definition and Notation

In multivariable calculus, a vector function describes a curve in space by assigning a vector to each value of a parameter, typically time t. For example,

• Position vector: $\vec{r}(t) = (f(t), g(t), h(t))$

• Velocity vector: $\vec{v}(t) = \frac{d\vec{r}}{dt}$

• Acceleration vector: $\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$

These vectors describe the motion of a particle along a path in three-dimensional space.

Tangent Lines to Vector Curves

Equation of the Tangent Line

The tangent line to a curve at a point gives the direction in which the curve is heading at that instant. For a vector function $\vec{r}(t)$ at $t = t_0$:

• The tangent line at $t_0$ is given by: $\vec{L}(t) = \vec{r}(t_0) + s\vec{v}(t_0)$, where $s$ is a real parameter.

• Example: If $\vec{r}(t) = (t-3)\vec{i} + t\vec{j} + t^2\vec{k}$, then $\vec{v}(t) = \frac{d\vec{r}}{dt} = \vec{i} + \vec{j} + 2t\vec{k}$.

Integration of Vector Functions

Definite Integrals

The definite integral of a vector function over an interval $[a, b]$ is computed component-wise:

• $\int_a^b \vec{r}(t) dt = \left( \int_a^b f(t) dt, \int_a^b g(t) dt, \int_a^b h(t) dt \right)$

• This can represent, for example, the net displacement of a particle over the interval.

Curvature of a Space Curve

Definition and Formula

Curvature measures how quickly a curve changes direction at a given point. For a vector function $\vec{r}(t)$:

• The curvature $\kappa(t)$ is given by:

$\kappa(t) = \frac{|\vec{v}(t) \times \vec{a}(t)|}{|\vec{v}(t)|^3}$

• Where $\vec{v}(t) = \frac{d\vec{r}}{dt}$ and $\vec{a}(t) = \frac{d^2\vec{r}}{dt^2}$.

• To find the curvature at a specific point, substitute the value of $t$ corresponding to that point.

Recovering Initial Conditions from Acceleration

Finding Original Velocity and Position Vectors

If the acceleration vector $\vec{a}(t)$ is known, and the position and velocity at a specific time $t_1$ are given, the original velocity $\vec{v}(0)$ and position $\vec{r}(0)$ can be found by integrating:

• $\vec{v}(t) = \int \vec{a}(t) dt + \vec{C}_1$

• $\vec{r}(t) = \int \vec{v}(t) dt + \vec{C}_2$

• Use the given values at $t_1$ to solve for the constants $\vec{C}_1$ and $\vec{C}_2$.

Arc Length of a Curve

Formula for Arc Length

The arc length $L$ of a curve $\vec{r}(t)$ from $t = a$ to $t = b$ is:

$L = \int_a^b |\vec{v}(t)| dt$

• Where $|\vec{v}(t)|$ is the magnitude of the velocity vector.

• This gives the total distance traveled along the curve.

Unit Tangent and Principal Unit Normal Vectors

Definitions and Formulas

• Unit tangent vector: $\vec{T}(t) = \frac{\vec{v}(t)}{|\vec{v}(t)|}$

• Principal unit normal vector: $\vec{N}(t) = \frac{d\vec{T}/dt}{|d\vec{T}/dt|}$

• These vectors describe the direction of motion and the direction in which the curve is turning, respectively.

Components of Acceleration: Tangential and Normal

Decomposition of Acceleration

The acceleration vector $\vec{a}(t)$ can be decomposed into tangential and normal components:

$\vec{a}(t) = a_T \vec{T} + a_N \vec{N}$

• Tangential component: $a_T = \frac{d|\vec{v}|}{dt}$ (rate of change of speed)

• Normal component: $a_N = \kappa |\vec{v}|^2$ (change in direction)

Reference Table: Key Vector Calculus Quantities

Quantity	Formula
Velocity	$\vec{v}(t) = \frac{d\vec{r}}{dt}$
Acceleration	$\vec{a}(t) = \frac{d\vec{v}}{dt}$
Unit Tangent Vector	$\vec{T}(t) = \frac{\vec{v}(t)}{|\vec{v}(t)|}$
Principal Unit Normal Vector	$\vec{N}(t) = \frac{d\vec{T}/dt}{|d\vec{T}/dt|}$
Curvature	$\kappa = \frac{|\vec{v} \times \vec{a}|}{|\vec{v}|^3}$
Arc Length	$L = \int_a^b |\vec{v}(t)| dt$
Acceleration Decomposition	$\vec{a}(t) = a_T \vec{T} + a_N \vec{N}$

Example Application

• Given $\vec{r}(t) = (\cos(6t), \sin(6t), t-6)$, to find arc length from $t=0$ to $t=3$, compute $|\vec{v}(t)|$ and integrate over the interval.

• To find the curvature at a point, use the velocity and acceleration vectors at that $t$.

Additional info: The above notes synthesize the main concepts and formulas required to answer the provided questions, including vector differentiation, tangent lines, curvature, arc length, and the decomposition of acceleration. These are core topics in a multivariable calculus (Calculus III) course.