Area Vectors

Section 22.2 Area Vectors

Exercises Warm-up Activities

1.

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?

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The dot product was introduced in The Dot Product.

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To describe surfaces, we will need to define a new quantity.

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Definition 22.2.1. 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 \(\mathrm{m}^2\)

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Figure 22.2.2. An example of the area vector.
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Area vectors will help us define different surfaces, and will become particularly useful in the next few sections.

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Exercises Activities

Below is a picture of a closed surface (the outside cover of a book).

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Figure 22.2.3. A simple representation of a book.
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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?

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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!

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