Math 251 Section 11.6 Vector Functions and Space Curves
A curve described by C:

r ( t )  x ( t ), y ( t ), z ( t )
 x (t ), y (t ), z (t ))  : a  t  b has vector equation
atb

Limits and derivatives are taken componentwise. r (t ) is continuous if and only if each of
the component functions is continuous.

Examples:



Sketch the curves: 1. r (t )  (cos t ) i  (sin t ) j  t k




2. r ( t )  ( 3 cos t ) i  ( 4 sin t ) j  t k




3. r ( t )  ( 3 sin t ) i  ( 4 cos t ) j  t k
Derivatives and integrals are done component-wise. See Theorem 5 in the box on pg 694
in Stewart.
1 and 2 together make the linear property.




 
 a u  b v  ( t )  a u ( t )  b v ( t )


Note the 3 product rules:
I






 f ( t ) r ( t )   f ( t ) r ( t )  f ( t ) r ' ( t )


Ex. Find
II
d
dt



 t
 e  (cos t ) i  (sin t ) j  t k  

 

 
 






 s ( t )  r ( t )   s ( t )  r ( t )  s ( t )  r ( t )


d
Ex. Find
dt
III


1  
  2  3   
 t i  t j  t k    ln t i  2t j  t 2 k  
 


 






 s ( t )  r ( t )   s ( t )  r ( t )  s ( t )  r ( t )


Note: The order must be
maintained.
d
Ex. Find
dt


1  
  2  3   
 t i  t j  t k    ln t i  2t j  t 2 k  
 


The Chain Rule





 r ( g (t )   g (t ) r ( g (t ))


Ex. Find




d  2 
 r ( t )  for r ( t )  ( 3 sin t ) i  ( 4 cos t ) j  t k .
dt 


For a curve with vector equation r ( t )  x ( t ), y ( t ), z ( t )
a  t  b,


T (t ) 
the Unit Tangent Vector is defined as
r ( t )

r ( t )
.
Ex. Find the unit tangent vector function




for r (t )  (cos t ) i  (sin t ) j  t k .


Are r (t ) and T (t ) orthogonal?


Ex. Show that r (t ) and T (t ) are orthogonal if and only if the curve lies on a sphere.
Integrals are done componentwise.
1
Ex. Evaluate

te t , e 2 t , e 3t dt
0
Examples from Stewart, pg 697


#42. Given r (t )  e 2 t , e  2 t , te 2 t , find the unit tangent vector T ( 0) .
 

#44. Given r (t )  e 2 t cos t , e 2 t sin t , e 2 t , find the unit tangent vector, T ( ) .
2
49 and 50 pg 607 Find parametric equations for the tangent line to the given curve at the
given point.
49 .
x  t,
y
50 .
x  cos t ,
2 cos t ,
y  3e 2 t ,
4
60 pg 698 Evaluate
z
2 sin t
z  3e  2 t
(

4
,1,1)
(1,3,3)

 
1 
 t i  te  t j 
k  dt
2
t


1
