The document discusses representations of lines, planes, and space curves in vector calculus. It defines the vector equation of a line, parametric equation of a line, and symmetric equation of a line. It also defines the vector equation and scalar equation of a plane. It then discusses representing space curves using parametric vector functions and calculating derivatives and integrals of vector functions, including rules for differentiation. It provides an example problem proving that the tangent line to a circle at a point is perpendicular to the line connecting that point to the circle's center.

1. Vector equation of a line:
= +
If = , , , = , , , = , , ,
then
, , = + , + , +
Parametric equation:
= + , = + , = +
symmetric equation:
= =

2. Vector equation of a plane:
·( )=
If = , , , = , , , = , , ,
then:
, , · , , =
Scalar equation:
( )+ ( )+ ( )=
Linear equation:
+ + + =

3. Distance from a point ( , , ) to a plane
+ + + = is given by:
| + + + |
=
+ +
Ex: Find the distance between ( , , ) and
+ = .
Ex: Find the distance between the two planes
+ = , + = .

4. 13.1 Vector Functions
and Space Curves
A 3D space curve can be represented by the
parametric equations
= ( ), = ( ), = ()
It can be written as a vector function:
()= ( ), ( ), ( ) = ( ) + ( ) + ( )
Ex:
()= , ( ),

5. Ex: ()= , ,

6. Ex: ()= ( + ) ,( + ) ,

7. Ex: ()= ( + . ) ,( + . ) , .

8. 13.2 Derivatives and
Integrals of Vec. Func.
Definition of the derivative of a vector function:
( + ) ()
()= ()=
=
If
()= ( ), ( ), ( ) = ( ) + ( ) + ( )
then
()= ( ), ( ), ( ) = ( ) + () + ()

9. Differentiation Rules:
( ( ) + ( )) = ( )+ ( )
( ( )) = ()
( ( ) ( )) = ( ) ( ) + ( ) ( )
( ( ) · ( )) = ( )· ( )+ ( )· ( )
( () ( )) = () ( )+ ( ) ()
( ( ( ))) = ( ) ( ( ))

10. Ex: Prove that the tangent line of a circle at
a point is perpendicular to the connecting
line of the point and the center.

11. Ex: Prove that the tangent line of a circle at
a point is perpendicular to the connecting
line of the point and the center.
A circle can be represented as | ( )| =

12. Ex: Prove that the tangent line of a circle at
a point is perpendicular to the connecting
line of the point and the center.
A circle can be represented as | ( )| =
We want to show that () ()

13. Ex: Prove that the tangent line of a circle at
a point is perpendicular to the connecting
line of the point and the center.
A circle can be represented as | ( )| =
We want to show that () ()
( ) · ( ) = | ( )| =

14. Ex: Prove that the tangent line of a circle at
a point is perpendicular to the connecting
line of the point and the center.
A circle can be represented as | ( )| =
We want to show that () ()
( ) · ( ) = | ( )| =
[ ( ) · ( )] =

15. Ex: Prove that the tangent line of a circle at
a point is perpendicular to the connecting
line of the point and the center.
A circle can be represented as | ( )| =
We want to show that () ()
( ) · ( ) = | ( )| =
[ ( ) · ( )] =
( )· ( )=

16. Ex: Prove that the tangent line of a circle at
a point is perpendicular to the connecting
line of the point and the center.
A circle can be represented as | ( )| =
We want to show that () ()
( ) · ( ) = | ( )| =
[ ( ) · ( )] =
( )· ( )=
() ()

17. Definite integral:
() = () , () , ()
The Fundamental Theorem of Calculus:
If ( ) = ( ), then
() = () = ( ) ( )