## 17Calculus Vector Functions - Equations

##### 17Calculus

This page contains a 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

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

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

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 - When 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

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

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

$$a_{\vhat{N}} = \vec{a} \cdot \vhat{N}$$

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

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

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

curvature

$$\displaystyle{ K(t) = \frac{1}{\|\vec{v}\|} \left\| \frac{d\vec{T}}{dt} \right\| = \frac{\| \vec{T}'(t) \|}{\|\vec{r}'(t)\|} }$$

$$\displaystyle{ K=\frac{\|\vec{v} \times \vec{a} \|}{\|\vec{v}\|^3} }$$

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

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