Chapter 21
Arrow Addition
The terms arrow and vector will be used interchangeably.
This applet (online only) illustrates the ideas in this chapter.
Map Addition Method
In navigation, drawing an arrow on a map from a point A to a point
B represents a linear
FOOTNOTE: Physically, the arrow from A to C should lie on the path a taut line or string would follow between the two points.displacement or movement between them, that is the tail point A and the head point B. To show a second displacement from the head point B, put the tail of one second arrow at B. The result of these two movements, is a nonlinear movement from a tail point A of the first arrow to the head point, say 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 called the sum of the first two. It represents a linear movement from the points A to the point C. The foregoing describes the head to tail map addition method for adding two arrows together when the head of one is at the tail of another.
Two of the solid arrows and dotted lines parallel to them in the above diagram form the sides and diagonal of a parallelogram. The three solid arrows form a triangle. Another triangle is formed by the dotted lines and the diagonal arrow of the parallelogram. The rotation of these triangles and the parallelogram will have a deep consequence in the following overlapping discussion of complex, sines and cosines.
Reading Guide
Too much may be said in this chapter. If you get lost in the details, read this chapter lightly or go on to the next chapter. One aim of this chapter was to make fuss about some technical details - gaps in the author's comprehension perhaps.
Parallelogram Addition Method
When two arrows have the same tail points, they determine a parallelogram as well. They can be added by moving without a change in direction, one the arrows to the head of the other. This gives the parallelogram method for adding or summing two arrows with the same tail points (or origins).
Observe the presence of a two triangles and a parallelogram.
The dotted lines indicate positions of two of the solid arrows, after a movement to the head of the other without a change of direction. We will describe each displacement as a parallel movement of one along the other. The arrows before and after movement altogether form the sides of a parallelogram. The addition of one arrow to a second is represented by the parallel movement of the tail point of one to the head of the other. The formation of the parallelogram implies that which is added to which is immaterial, the result will be the same. Either way, the solid arrow along the diagonal of the parallelogram gives the (linear) arrow sum of the other two.
Arrow Components
The above diagram shows how the arrow from A to B can be regarded as the parallelogram sum of a horizontal arrow and a vertical arrow. The horizontal and vertical arrows are respectively called the horizontal and vertical components of the initial arrow from A to B. (These components depend on the choice of directions for the so-called horizontal and vertical axes.) The initial arrow is the map and parallelogram sum of the two component arrows. In the representation of arrows, an arrow can be viewed as the map addition of its vertical component to horizontal component arrows. Here the tail of the vertical component is moved to the head of the horizontal component. (The arrow could be also be viewed as the map addition of the horizontal component to the vertical one.) Which map addition is shown on a diagram is immaterial.
Component Addition Method
The following diagram shows the map addition of two solid arrows, namely the tail to head addition of the arrow from B to C to the arrow from A to B gives the same result as parallelogram addition of (i) the sum of the vertical components to (ii) the sum of the horizontal components.
Why Slopes
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If you like Volume 3 you may also like Three Skills for Algebra , Exponents & Radicals Exactly, complex numbers, Euclidean Geometry , More Calculus and Pattern Based Reason as well. |
Units in Calculations:
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.