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# Calculus II - 33

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Stewart Calculus 12.3&4

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• ### Calculus II - 33

1. 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. 2. Properties (I): · = · ·( + )= · + ·( )· = ( · )= ·( ) · =
3. 3. Properties (II): · =| | · = | || | · =
4. 4. Properties (II): · =| | · = | || | · = · > · = · <
5. 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. 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. 7. 12.4 The Cross Product The cross product of = , , and = , , is given by = , , = = + It can only be defined for 3D vectors.
8. 8. Properties (I): = ( + )= +( ) = ( )= ( ) = · = · · )= · + ·( + ·( ) = ( · )= ( )· · =
9. 9. Properties (II): =| | = | || | = · =| | · = | || | · =
10. 10. Properties (III):( ) ,( )| | = | || | equals to the area ofthe parallelogram determined by and .
11. 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 .