You want to find the dot product between two vectors, \(\vec{A}\) and \(\vec{B}\text{.}\) What do you know about the directions of vectors if the dot product is positive? What if the dot product is negative? What if the dot product is zero?
To describe surfaces, we will need to define a new quantity.
Definition22.2.1.Area Vector.
The area vector \(\vec{A}\) for a surface points perpendicular to the surface, with a magnitude equal to the area of the surface. For an infinitesimal surface, the area vector is typically written \(d\vec{A}\text{.}\) An area vectors has units of \(\mathrm{m}^2\)
Figure22.2.2.An example of the area vector.
Area vectors will help us define different surfaces, and will become particularly useful in the next few sections.
ExercisesActivities
Below is a picture of a closed surface (the outside cover of a book).
Figure22.2.3.A simple representation of a book.
1.Area Vectors for a Book.
Which direction is the area vector for the front cover of the book? For the spine? How many different area vectors do you need for the entire book?
2.
Write a differential area vector \(d\vec{A}\) for a small part of the front cover of the book in terms of differential elements like \(dx\) and \(dy\text{.}\) Check that your expression has the correct units!
