Vector Valued Functions

Vector Valued Functions

BY DR. RAHIFA

Tangents, Normals and Binormals

• Tangent vector,  $T$  is a vector that is tangent to a curve or surface at a given point.  

• Normal vector,  $N$  is a vector which is perpendicular to the surface at a given point.

• Binormal vector,  $B$  is a vector that is orthogonal to both the tangent vector and the normal vector

• Curvature,  $\kappa$  the magnitude of the rate of change of the unit tangent with respect to arc length along the curve

• Torsion, $\tau$  the rate of change of the direction of the unit binormal vector.

Tangents, Normals and Binormals

FORMULAS

•  UNIT TANGENT VECTOR $\overline{T}=\frac{\overline{r}'}{\left|\overline{r}'\right|}=\frac{\overline{v}}{\left|\overline{v}\right|}$  

• UNIT NORMAL VECTOR  $\overline{N}=\frac{\overline{T}'}{\left|\overline{T}'\right|}$  

• CURVATURE,  $\kappa=\frac{\left|\overline{T}'\right|}{\left|\overline{r}'\right|}$  

FORMULAS

•  BINORMAL VECTOR $\overline{B}=\overline{T}\times\overline{N}$ 

• TORSION,  $\tau=\frac{\left|\overline{B}'\right|}{\left|\overline{r}'\right|}$  

EXAMPLE 1

Find the unit tangent vector, unit normal vector, curvature, binormal vector and torsion at each point on the graph of a vector function

 $\overline{r}\left(t\right)=\cos t\ \overline{i}+\sin t\ \overline{j}\ +t\ \overline{k}$  

EXAMPLE 1: Unit Tangent Vector

 $\overline{r}'\left(t\right)=-\sin t\ \overline{i}+\cos t\ \overline{j}\ +\ \overline{k}$  

  $\left|\overline{r}'\right|=\sqrt{\left(-\sin t\right)^2+\left(\cos t\right)^2+1}=\sqrt{\sin^2t+\cos^2t+1}=\sqrt{2}$ Unit Tangent Vector  $\overline{T}=\frac{\overline{r}'}{\left|\overline{r}'\right|}=\frac{-\sin t}{\sqrt{2}}\overline{i}+\frac{\cos t}{\sqrt{2}}\overline{j}+\frac{1}{\sqrt{2}}\overline{k}$  

EXAMPLE 1: Unit Normal Vector

 $\overline{T}'\left(t\right)=\frac{1}{\sqrt{2}}\left(-\cos t\ \overline{i}-\sin t\ \overline{j}\ \right)$  

  $\left|\overline{T}'\right|=\sqrt{\frac{\left(-\cos t\right)^2+\left(-\sin t\right)^2}{2}}=\sqrt{\frac{\cos^2t+\sin^2t}{2}}=\frac{1}{\sqrt{2}}$ 

Unit Normal Vector  $\overline{N}=\frac{\overline{T}'}{\left|T'\right|}=\frac{\frac{-\cos t}{\sqrt{2}}\overline{i}-\frac{\sin t}{\sqrt{2}}\overline{j}}{\frac{1}{\sqrt{2}}}=-\cos t\ \overline{i}-\sin t\ \overline{j}$  

EXAMPLE 1: Curvature

Curvature  $\kappa=\frac{\left|\overline{T}'\right|}{\left|\overline{r}'\right|}=\frac{\frac{1}{\sqrt{2}}}{\sqrt{2}}=\frac{1}{2}$  

EXAMPLE 1: Binormal Vector

Unit Binormal Vector   
 $\overline{B}=\overline{T\ }\times\overline{N}$ 

 

EXAMPLE 1: Torsion

 $\overline{B}'\left(t\right)=\frac{1}{\sqrt{2}}\left(\cos t\ \overline{i}\ +\sin t\ \overline{j}\right)$  

 $\left|\overline{B}'\right|=\sqrt{\frac{\cos^2t+\sin^2t}{2}}=\frac{1}{\sqrt{2}}$  

Torsion $\tau=\frac{\left|\overline{B}\right|}{\left|\overline{r}'\right|}=\frac{\frac{1}{\sqrt{2}}}{\sqrt{2}}=\frac{1}{2}$  

Multiple Choice

Exercise 1:
Given a vector function

 $\overline{r}\left(t\right)=3\sin t\ \overline{i}\ +3\ \cos t\ \overline{j}\$  

1. Find the derivative,  $\overline{r}'$  and its magnitude $\left|\overline{r}'\right|$ . 

•  $\overline{r}'=3\cos t\ \overline{i}\ -3\sin t\ \overline{j},\ \ \ \left|\overline{r}'\right|=3$  

•  $\overline{r}'=3\cos t\ \overline{i}\ +3\sin t\ \overline{j},\ \ \ \left|\overline{r}'\right|=3$  

•  $\overline{r}'=-3\cos t\ \overline{i}\ -3\sin t\ \overline{j},\ \ \ \left|\overline{r}'\right|=3$  

•  $\overline{r}'=-3\cos t\ \overline{i}\ +3\sin t\ \overline{j},\ \ \ \left|\overline{r}'\right|=3$  

Multiple Choice

Exercise 1:
Given a vector function

 $\overline{r}\left(t\right)=3\sin t\ \overline{i}\ +3\ \cos t\ \overline{j}\ .$  

2. Find the unit tangent vector  $\overline{T}$ 

•  $\overline{T}=\cos t\ \overline{i}\ -\sin t\ \overline{j}$  

•  $\overline{T}=-\cos t\ \overline{i}\ -\sin t\ \overline{j}$  

•  $\overline{T}=\cos t\ \overline{i}\ +\sin t\ \overline{j}$  

•  $\overline{T}=-\cos t\ \overline{i}\ +\sin t\ \overline{j}$  

Multiple Choice

Exercise 1:
Given a vector function

 $\overline{r}\left(t\right)=3\sin t\ \overline{i}\ +3\ \cos t\ \overline{j}\$  

3. Find the derivative,  $\overline{T}'$  and its magnitude $\left|\overline{T}'\right|$ . 

•  $\overline{T}'=-\sin t\ \overline{i}\ -\cos t\ \overline{j},\ \ \ \left|\overline{T}'\right|=1$  

•  $\overline{T}'=\sin t\ \overline{i}\ -\cos t\ \overline{j},\ \ \ \left|\overline{T}'\right|=1$  

•  $\overline{T}'=-\sin t\ \overline{i}\ +\cos t\ \overline{j},\ \ \ \left|\overline{T}'\right|=1$  

•  $\overline{T}'=\sin t\ \overline{i}\ +\cos t\ \overline{j},\ \ \ \left|\overline{T}'\right|=1$  

Fill in the Blanks

Exercise 1:
Given a vector function

 $\overline{r}\left(t\right)=3\sin t\ \overline{i}\ +3\ \cos t\ \overline{j}\ .$  

4. Find the curvature  $\kappa$ 

Multiple Choice

Exercise 1:
Given a vector function

 $\overline{r}\left(t\right)=3\sin t\ \overline{i}\ +3\ \cos t\ \overline{j}\$  

5. Find the unit normal vector,  $\overline{N}$  . 

•  $\overline{N}=-\ \sin t\ \overline{i}\ -\cos t\ \overline{j}\$  

•  $\overline{N}=\ \sin t\ \overline{i}\ -\cos t\ \overline{j}\$  

•  $\overline{N}=-\ \sin t\ \overline{i}\ +\cos t\ \overline{j}\$  

•  $\overline{N}=\ \sin t\ \overline{i}\ +\cos t\ \overline{j}\$  

Multiple Choice

Exercise 1:
Given a vector function

 $\overline{r}\left(t\right)=3\sin t\ \overline{i}\ +3\ \cos t\ \overline{j}\$  

6. Find the binormal vector,  $\overline{B}=\overline{T}\times\overline{N}$  . 

•  $\overline{B}=-\ \overline{k}$  

•  $\overline{B}=\ \overline{k}$  

•  $\overline{B}=2\ \overline{k}$  

•  $\overline{B}=-2\ \overline{k}$  

Fill in the Blanks

Exercise 1:
Given a vector function

 $\overline{r}\left(t\right)=3\sin t\ \overline{i}\ +3\ \cos t\ \overline{j}\ .$  

7. Find the torsion $\tau$ 

THE END