Stewart Calculus 12.4&5

1. 12.4 The Cross Product
The cross product of = , , and
= , , is given by
= , ,
=
= +
It can only be defined for 3D vectors.

2. Properties (I):
=
( + )= +
( ) = ( )= ( )
=
· = ·
·
)= · +
·( +
·( )
= ( · )=
( )·
· =

3. Properties (II):
=
| | = | || |
=
· =| |
· = | || |
· =

4. 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 .

5. Properties (IV):
·( )=( )·
·( ) is called the scalar triple product
of , , .
·( )=
The volume of the parallelepiped
determined by the vectors
, , equals | ·( )|.

6. Properties (V):
( )=( · ) ( · )
( )=( )

7. Properties (V):
( )=( · ) ( · )
( )=( )
Properties (I-IV):
=
( + )= +
( ) = ( )= ( )
=
=
| | = | || |
=
( ) ,( )
·( )=( )·

8. 12.5 Equations of Lines
and Planes
Vector equation of a line:
= +
If = , , , = , , , = , , ,
then
, , = + , + , +
Parametric equation:
= + , = + , = +

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

10. Ex: Find an equation of the line pass through
two given points ( , , ) and ( , , )
Ex: Show that the lines with parametric
equations
= + , = + , =
= , = + , = +
do not intersect and are not parallel.

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

12. Ex: Find an equation of the plane through
the point ( , , ) with normal vector , ,
Ex: Find an equation of the plane that passes
through ( , , ), ( , , ) and ( , , ).
Ex: Find the point at which the line with
parametric equations
= + , = + , =
intersects the plane
+ = .
Ex: Find a equation for the line of
intersection of two planes
= , + = .