Diese Präsentation wurde erfolgreich gemeldet.
Wir verwenden Ihre LinkedIn Profilangaben und Informationen zu Ihren Aktivitäten, um Anzeigen zu personalisieren und Ihnen relevantere Inhalte anzuzeigen. Sie können Ihre Anzeigeneinstellungen jederzeit ändern.
Nächste SlideShare
Wird geladen in …5
×

# Calculus II - 34

819 Aufrufe

Veröffentlicht am

Stewart Calculus 12.4&5

Veröffentlicht in: Technologie, Bildung
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Als Erste(r) kommentieren

• Gehören Sie zu den Ersten, denen das gefällt!

### Calculus II - 34

1. 1. 12.4 The Cross Product The cross product of = , , and = , , is given by = , , = = + It can only be defined for 3D vectors.
2. 2. Properties (I): = ( + )= +( ) = ( )= ( ) = · = · · )= · + ·( + ·( ) = ( · )= ( )· · =
3. 3. Properties (II): =| | = | || | = · =| | · = | || | · =
4. 4. Properties (III):( ) ,( )| | = | || | equals to the area ofthe parallelogram determined by and . The Right Hand Rule: If the fingers of your right hand curl in the direction of a rotation from to , then your thumb points in the direction of .
5. 5. Properties (IV): ·( )=( )· ·( ) is called the scalar triple product of , , . ·( )=The volume of the parallelepipeddetermined by the vectors , , equals | ·( )|.
6. 6. Properties (V): ( )=( · ) ( · ) ( )=( )
7. 7. Properties (V): ( )=( · ) ( · ) ( )=( )Properties (I-IV): = ( + )= +( ) = ( )= ( ) = =| | = | || | =( ) ,( ) ·( )=( )·
8. 8. 12.5 Equations of Lines and Planes Vector equation of a line: = + If = , , , = , , , = , , , then , , = + , + , + Parametric equation: = + , = + , = +
9. 9. Vector equation of a line: = +If = , , , = , , , = , , ,then , , = + , + , +Parametric equation: = + , = + , = +symmetric equation: = =
10. 10. Ex: Find an equation of the line pass throughtwo given points ( , , ) and ( , , )Ex: Show that the lines with parametricequations = + , = + , = = , = + , = +do not intersect and are not parallel.
11. 11. Vector equation of a plane: ·( )=If = , , , = , , , = , , ,then: , , · , , =Scalar equation: ( )+ ( )+ ( )=Linear equation: + + + =
12. 12. Ex: Find an equation of the plane throughthe point ( , , ) with normal vector , ,Ex: Find an equation of the plane that passesthrough ( , , ), ( , , ) and ( , , ).Ex: Find the point at which the line withparametric equations = + , = + , =intersects the plane + = .Ex: Find a equation for the line ofintersection of two planes = , + = .