Arrows, Vectors and Complex Numbers
Chapter 4:
In geometry and other parts of mathematics, arrows are called vectors. Arrows and vectors have both directions and magnitudes (or lengths). For instance, the information that a restaurant with excellent food and coffee is 100 steps due north of your present location defines a length and direction to a fine dining area. This vectorial information, the length and direction, is most appealing when you are hungry and you have sufficient funds in your pocket. This vectorial information indicates a possible displacement or movement.
Arrows or vectors, that is lengths with a direction for movement, play a role in navigation. The small print below shows why, and is optional reading.
On a map, drawing an arrow from a point A to a point B represents a linear (straight) displacement or movement between them, the tail point A and the head point B. To show a second displacement from the head point B to another point C, draw a second arrow from B to C. This second arrow with tail at B and head at C represents a second movement.This small print describes the head to tail method for arrow and vector addition. The two words arrow and vector are interchangeable. While the latter is more proper, the former will be favoured in this work.
The two arrows together show a nonlinear movement from a tail point A of the first arrow to the head point C of the second arrow. The straight arrow joining the tail point A of the first arrow to the head point C of the second is a third arrow. It is called the sum of the first two.
The description of the multiplication of arrows in the coordinate plane
follows in small print - more optional reading. The add the angles, multiply the
lengths rule in this small print implies a simplification for exposition of both
complex numbers and trigonometry.
A coordinate plane has two axes, one is called horizontal and the other vertical. The intersection of the two axes is called the origin. If the tail of an arrow is place at the origin, it forms an angle with the horizontal axis. The size and direction of the arrow is given, that is determined by its length and this angle. Feynman's repeated instruction to obtain or compute the product of two arrows was as follows: add their angles and multiply their lengths to obtain the angle and length of the arrow which is their product. Note the absence of units of length here.
In this example, the angle of the product is given by the sum of the angles of the factors, that is 69.59° = 22.62°+46.97° while the product of the factor's lengths (1.026)(1.3) = 1.3338 gives the length of the product.
The rule add the angles, multiply the lengths is the polar coordinate method for multiplication. A knowledge of the how to multiply nonnegative real numbers and of how rectangular (Cartesian) and polar coordinates locate points in the plane is enough to understand it. So this multiplication could be explained to students who have yet to master (i) the law of signs for real number multiplication and/or (ii) the algebraic way of writing and thinking. Indeed with the identification of the horizontal axis in the coordinate plane with real numbers, the rule add the angle, multiply the lengths can be used to imply or confirm the law of signs. This add the angles, multiply the lengths rule can be explained and understood before multiplication by negative numbers is explained. The latter multiplication can even be introduced by this rule. See the chapters on complex numbers (basic ideas) in the companion book Why Slopes and More Math for more details.
Remark. In the companion book, the chapters on complex numbers show in two different ways how the add the angles and multiply the lengths rule leads to the standard expressions for the rectangular coordinates, alias real and imaginary parts, of a product. Here two ways imply (and do not depend on) the angle sum formulas in trigonometry. Moreover, add the angles and multiply the lengths rule along with the standard expressions justifies the use of complex exponential, and so immediately simplifies the treatment of trig in a manner familiar to students of engineering and physics.
Remark. If in the polar coordinates (R,q),
the distance R = r ·U where U denotes a unit of length then
defining the product as
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