Multivariable Calculus
Lecture 2
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
President University
Erwin Sitompul
MVC 2/1
Chapter 12
12.4 The Cross Product
The Cross Product of Two Vectors in Space
 In space, we need a way to describe
how a plane is tilting. We accomplish
this by multiplying two vectors in the
plane together to get a third vector
perpendicular to the plane
 The direction of this third vector tells
us the “inclination” of the plane.
 We use cross product to multiply the
vectors together.
President University
Erwin Sitompul
MVC 2/2
Chapter 12
12.4 The Cross Product
The Cross Product of Two Vectors in Space
President University
Erwin Sitompul
MVC 2/3
Chapter 12
12.4 The Cross Product
The Cross Product of Two Vectors in Space
President University
Erwin Sitompul
MVC 2/4
Chapter 12
12.4 The Cross Product
The Cross Product of Two Vectors in Space
 Example
President University
Erwin Sitompul
MVC 2/5
Chapter 12
12.4 The Cross Product
|u  v| is the Area of a Parallelogram
President University
Erwin Sitompul
MVC 2/6
Chapter 12
12.4 The Cross Product
Distance and Spheres in Space
 Example
 Example
President University
Erwin Sitompul
MVC 2/7
Chapter 12
12.5 Lines and Planes in Space
Lines in Space
 Suppose L is a line in space passing
through a point P0(x0,y0,z0) parallel
to a vector v.
 Then L is the set of all points
P(x,y,z) for which P0P is parallel to
v.
 P0P = tv, for a given value of scalar
parameter t.
President University
Erwin Sitompul
MVC 2/8
Chapter 12
12.5 Lines and Planes in Space
Lines in Space
President University
Erwin Sitompul
MVC 2/9
Chapter 12
12.5 Lines and Planes in Space
Lines in Space
 Example
President University
Erwin Sitompul
MVC 2/10
Chapter 12
12.5 Lines and Planes in Space
Lines in Space
 Example
 What if we choose Q(1,–1,4) as the base?
President University
Erwin Sitompul
MVC 2/11
Chapter 12
12.5 Lines and Planes in Space
The Distance from a Point to a Line in Space
President University
Erwin Sitompul
MVC 2/12
Chapter 12
12.5 Lines and Planes in Space
The Distance from a Point to a Line in Space
 Example
President University
Erwin Sitompul
MVC 2/13
Chapter 12
12.5 Lines and Planes in Space
The Distance from a Point to a Plane
President University
Erwin Sitompul
MVC 2/14
Chapter 12
12.5 Lines and Planes in Space
The Distance from a Point to a Plane
 Example
President University
Erwin Sitompul
MVC 2/15
Chapter 13
Vector-Valued Functions and
Motion in Space
President University
Erwin Sitompul
MVC 2/16
Chapter 13
13.1 Vector Functions
Vector Functions
President University
Erwin Sitompul
MVC 2/17
Chapter 13
13.1 Vector Functions
Vector Functions
 Can you see the difference?
President University
Erwin Sitompul
MVC 2/18
Chapter 13
13.1 Vector Functions
Vector Functions
President University
Erwin Sitompul
MVC 2/19
Chapter 13
13.1 Vector Functions
Limits and Continuity
President University
Erwin Sitompul
MVC 2/20
Chapter 13
13.1 Vector Functions
Limits and Continuity
President University
Erwin Sitompul
MVC 2/21
Chapter 13
13.1 Vector Functions
Derivatives and Motion
President University
Erwin Sitompul
MVC 2/22
Chapter 13
13.1 Vector Functions
Derivatives and Motion
President University
Erwin Sitompul
MVC 2/23
Chapter 13
13.1 Vector Functions
Derivatives and Motion
 Example
President University
Erwin Sitompul
MVC 2/24
Chapter 13
13.1 Vector Functions
Derivatives and Motion
President University
Erwin Sitompul
MVC 2/25
Chapter 13
13.1 Vector Functions
Differentiation Rules
President University
Erwin Sitompul
MVC 2/26
Chapter 13
13.1 Vector Functions
Vector Functions of Constant Length
President University
Erwin Sitompul
MVC 2/27
Chapter 13
13.1 Vector Functions
Vector Functions of Constant Length
 Example
President University
Erwin Sitompul
MVC 2/28
Chapter 13
13.1 Vector Functions
Integrals of Vector Functions
 Example
President University
Erwin Sitompul
MVC 2/29
Chapter 13
13.1 Vector Functions
Integrals of Vector Functions
 Example
President University
Erwin Sitompul
MVC 2/30
Chapter 13
13.1 Vector Functions
Integrals of Vector Functions
 Example
President University
Erwin Sitompul
MVC 2/31
Chapter 13
13.1 Vector Functions
Integrals of Vector Functions
President University
Erwin Sitompul
MVC 2/32
Chapter 13
13.1 Vector Functions
Homework 2
 Exercise
 Exercise
 Exercise
 Exercise
 Exercise
 Exercise
12.4,
12.4,
12.5,
12.5,
13.1,
13.1,
No.
No.
No.
No.
No.
No.
15.
36.
6.
43.
7.
25.
 Due: Next week, at 17.15.
President University
Erwin Sitompul
MVC 2/33