### Key Terms

- acceleration vector
- the second derivative of the position vector

- arc-length function
- a function $s\left(t\right)$ that describes the arc length of curve
*C*as a function of*t*

- arc-length parameterization
- a reparameterization of a vector-valued function in which the parameter is equal to the arc length

- binormal vector
- a unit vector orthogonal to the unit tangent vector and the unit normal vector

- component functions
- the component functions of the vector-valued function $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$ are $f\left(t\right)$ and $g\left(t\right),$ and the component functions of the vector-valued function $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k}$ are $f\left(t\right),$ $g\left(t\right)$ and $h\left(t\right)$

- curvature
- the derivative of the unit tangent vector with respect to the arc-length parameter

- definite integral of a vector-valued function
- the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function

- derivative of a vector-valued function
- the derivative of a vector-valued function $\text{r}\left(t\right)$ is ${r}^{\prime}\left(t\right)=\underset{\text{\Delta}t\to 0}{\text{lim}}\frac{\text{r}\left(t+\text{\Delta}t\right)-\text{r}\left(t\right)}{\text{\Delta}t},$ provided the limit exists

- Frenet frame of reference
- (TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector

- helix
- a three-dimensional curve in the shape of a spiral

- indefinite integral of a vector-valued function
- a vector-valued function with a derivative that is equal to a given vector-valued function

- Kepler’s laws of planetary motion
- three laws governing the motion of planets, asteroids, and comets in orbit around the Sun

- limit of a vector-valued function
- a vector-valued function $\text{r}(t)$ has a limit
**L**as*t*approaches*a*if $\underset{t\to a}{\text{lim}}\left|\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)-\text{L}\right|=0$

- normal component of acceleration
- the coefficient of the unit normal vector
**N**when the acceleration vector is written as a linear combination of $\text{T}$ and $\text{N}$

- normal plane
- a plane that is perpendicular to a curve at any point on the curve

- osculating circle
- a circle that is tangent to a curve
*C*at a point*P*and that shares the same curvature

- osculating plane
- the plane determined by the unit tangent and the unit normal vector

- plane curve
- the set of ordered pairs $\left(f\left(t\right),g\left(t\right)\right)$ together with their defining parametric equations $x=f\left(t\right)$ and $y=g\left(t\right)$

- principal unit normal vector
- a vector orthogonal to the unit tangent vector, given by the formula $\frac{{T}^{\prime}\left(t\right)}{\Vert {T}^{\prime}\left(t\right)\Vert}$

- principal unit tangent vector
- a unit vector tangent to a curve
*C*

- projectile motion
- motion of an object with an initial velocity but no force acting on it other than gravity

- radius of curvature
- the reciprocal of the curvature

- reparameterization
- an alternative parameterization of a given vector-valued function

- smooth
- curves where the vector-valued function $\text{r}(t)$ is differentiable with a non-zero derivative

- space curve
- the set of ordered triples $\left(f\left(t\right),g\left(t\right),h\left(t\right)\right)$ together with their defining parametric equations $x=f\left(t\right),$ $y=g\left(t\right)$ and $z=h\left(t\right)$

- tangent vector
- to $\text{r}\left(t\right)$ at $t={t}_{0}$ any vector
**v**such that, when the tail of the vector is placed at point $\text{r}\left({t}_{0}\right)$ on the graph, vector**v**is tangent to curve*C*

- tangential component of acceleration
- the coefficient of the unit tangent vector
**T**when the acceleration vector is written as a linear combination of $\text{T}$ and $\text{N}$

- vector parameterization
- any representation of a plane or space curve using a vector-valued function

- vector-valued function
- a function of the form $\text{r}(t)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$ or $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k},$ where the component functions
*f, g,*and*h*are real-valued functions of the parameter*t*

- velocity vector
- the derivative of the position vector