The dot product of two vectors a and b is defined as a·b and is calculated by multiplying their corresponding components and adding them. It can be used to calculate the projection of one vector onto another and the cosine of the angle between the vectors. The cross product of two vectors a and b is defined as axb and produces a third vector that is perpendicular to both a and b. It has various properties including being anticommutative and having a magnitude equal to the area of the parallelogram determined by the two vectors.

1. 12.3 The Dot Product
The 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 Cosines
The direction angles of a nonzero vector are
the angles that it makes with the positive x-,
y-, and z-axes.
· ·
= = = =
| || | | | | || | | |
·
= =
| || | | |
+ + =
=| | , ,

6. Projections:
The scalar projection of onto (also called
the component of onto ) is defined to be
·
=| | = calar!
| | sig ne d s
The 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 of
the parallelogram determined by and .

11. Properties (III):
( ) ,( )
| | = | || | equals to the area of
the 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 .