Section 11.4 The Cross Product

1
Section 11.4The Cross Product

• Calculus III
• September 22, 2009
• Berkley High School

2
Operations on Vectors(So Far)

• Scalar Multiplication
• Vector Addition
• Vector Subtraction
• Dot Product

3
New Operation Cross Product
4
Definition of Determinant
5
Definition of Determinant
6
Definition of Cross Product
7
Example
8
Example
9
Properties Direction Magnitude
10
Properties Direction Magnitude
11
Properties Area of Parallelogram
12
Properties
13
Assignment

• Section 11.4, 1-35, odd x25
• But wait, theres more

14
Volume of Parallelepiped
15
Volume of Parallelepiped
16
Volume of Parallelepiped
17
Volume of Parallelepiped
18
Volume of Parallelepiped
19
Volume of Parallelepiped
20
Volume of Parallelepiped
Triple Scalar Product
21
Example
22
Example
23
Are two vectors coplanar?

• Any two non-zero vectors define a plane.
• Are three vectors coplanar?
• The vectors are coplanar iff the parallelepiped
  formed by the three vectors has Volume0

24
Example
25
Are three points coplanar?

• Any three points define a plane.
• Are four points coplanar?
• From the four points, three vectors can be formed
  with a common tail. If the vectors are coplanar,
  then the points are coplanar.

26
Assignment

• 11.4, 41-47 odd