Stewart 7th Edition ET
Calculus
12.4
Dr. Berg
The Cross Product
The cross product or vector product is a rather peculiar construction since it is
defined only for real three vectors, but it is a natural construction for things like torque.
Definition
If a

Stewart 7th Edition ET
Calculus
12.3
Dr. Berg
The Dot Product
Another useful operation between two vectors is the dot product. This gives the
vector space a Euclidean geometry.
Definition
The dot product, scalar product, or inner product of a = (a1,a2 ,a3

Stewart 7th Edition ET
Calculus
12.2
Dr. Berg
Vectors
Interpretation of Vectors
Vectors are commonly used to represent things that have magnitude and direction.
Motion of an object has magnitude (distance) and direction. Force has magnitude
(strength) and

Calculus
12.1
Stewart 7th Edition ET
Dr. Berg
Three-Dimensional Coordinate Systems
To locate a point in the plane, two numbers are necessary. Hence, we say that a
plane has two degrees of freedom, or is two-dimensional. By convention, a point with
coordin

Stewart 7th Edition ET
Calculus
10.4
Dr. Berg
Areas and Lengths in Polar Coordinates
Area
To develop a means of finding the area using polar coordinates, we begin by
observing that the area of a sector of a circle of fixed radius r is proportional to its

Stewart 7th Edition ET
Calculus
10.3
Dr. Berg
Polar Coordinates
The position of a point in a plane is determined by two numbers. We normally use
Cartesian coordinate systems, but there are others. Many curves are more easily described
using polar coordina

Stewart 7th Edition ET
Calculus
10.2
Dr. Berg
Calculus With Parametric Curves
Tangents
Suppose a curve is defined by the equation y = F(x) and can also be expressed as
the parametric equations x = f (t) and y = g(t) where F, f, and g are all differentiabl

Stewart 7th Edition ET
Calculus
10.1
Dr. Berg
Curves Defined by Parametric Equations
Parametric Equations
Some curves are most easily defined by giving the x and y coordinates as
functions of a third parameter t.
Definition
A parametric curve (in R2) is a