## 17Calculus Vector Functions - Equations

Using Vectors

Applications

### Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

Practice

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

### Articles

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

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

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

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 vectors vector functions projectile motion unit tangent vector principal unit normal vector acceleration components arc length arc length parameter curvature

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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