\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \)
\( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \)
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17calculus > vector functions > equations

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Vector Functions Equations

This page contains the full list of vector function equations including links to the pages where they are discussed and derived. Which equation you use will depend on the data you have to work with. We start by giving the variables and their names.

Variables

parametric equations

 

\(x(t),~y(t),~z(t)\)

position vector function

 

\(\vec{r}(t) = x(t)\vhat{i} + y(t)\vhat{j} + z(t)\vhat{k}\)

vector function in terms of the arc length parameter, s

 

\(\vec{r}(s) = x(s)\vhat{i} + y(s)\vhat{j} + z(s)\vhat{k}\)

Note - If the variable s is used on this page, it refers to the arc length parameter.

Equations

velocity

 

\(\vec{v}(t) = \vec{r}'(t)\)

basic acceleration

 

\(\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)\)

unit tangent vector

 

\(\displaystyle{ \vhat{T}(t) = \frac{\vec{r}'(t)}{ \| \vec{r}'(t) \| } }\)

 

\(\displaystyle{ \vhat{T}(t) = \frac{\vec{v}(t)}{ \| \vec{v}(t) \| } }\)

principal unit normal vector

 

\(\displaystyle{ \vhat{N}(t) = \frac{d\vhat{T}/dt}{ \| d\vhat{T}/dt \| } }\)

acceleration vector components

 

\( \vec{a}(t) = a_{\vhat{T}}\vhat{T} + a_{\vhat{N}}\vhat{N}\)

tangential component of acceleration

 

\(a_{\vhat{T}} = \vec{a} \cdot \vhat{T} \)

 

\(\displaystyle{a_{\vhat{T}} = \frac{\vec{a} \cdot \vec{v}}{\|\vec{v}\|} }\)

 

\(a_{\vhat{T}} = \| \vec{v} \|' \)

normal component of acceleration

 

\(a_{\vhat{N}} = \vec{a} \cdot \vhat{N} \)

 

\(a_{\vhat{N}} = \|\vec{v}\| \|\vhat{T}'\|\)

 

\(\displaystyle{a_{\vhat{N}} = \frac{\|\vec{v} \times \vec{a}\|}{\|\vec{v}\|} }\)

 

\(a_{\vhat{N}} = \sqrt{\|\vec{a}\|^2 - a_{\vhat{T}}^2}\)

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