Your SlideShare is downloading. ×
0
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Calculus II - 33
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Calculus II - 33

216

Published on

Stewart Calculus 12.3&4

Stewart Calculus 12.3&4

Published in: Technology, Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
216
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
6
Comments
0
Likes
1
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • Transcript

    • 1. 12.3 The Dot ProductThe dot product (or inner product) of = , , and = , , is given by · = + +It can also be defined for 2D vectors.Ex: , , · , , = · + ( )+ · =+ · = · + · +( )( )=
    • 2. Properties (I): · = · ·( + )= · + ·( )· = ( · )= ·( ) · =
    • 3. Properties (II): · =| | · = | || | · =
    • 4. Properties (II): · =| | · = | || | · = · > · = · <
    • 5. Direction Angles and Direction CosinesThe direction angles of a nonzero vector arethe angles that it makes with the positive x-,y-, and z-axes. · · = = = = | || | | | | || | | | · = = | || | | | + + = =| | , ,
    • 6. Projections:The scalar projection of onto (also calledthe component of onto ) is defined to be · =| | = calar! | | sig ne d sThe projection of onto is defined to be · · = = ve ctor! | | | | | |
    • 7. 12.4 The Cross Product The cross product of = , , and = , , is given by = , , = = + It can only be defined for 3D vectors.
    • 8. Properties (I): = ( + )= +( ) = ( )= ( ) = · = · · )= · + ·( + ·( ) = ( · )= ( )· · =
    • 9. Properties (II): =| | = | || | = · =| | · = | || | · =
    • 10. Properties (III):( ) ,( )| | = | || | equals to the area ofthe parallelogram determined by and .
    • 11. 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 .

    ×