Chapter 21. Arrow Addition

The terms arrow and vector will be used interchangeably.

Map Addition Method

In navigation, drawing an arrow on a map from a point A to a point B represents a linear
displacement or movement between them, that is the tail point A and the head point B.

FOOTNOTE: Physically, the arrow from A to C should lie on the path a taut line or string would follow between the two points.

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

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.

The following diagram show the parallelogram addition of two arrows, gives the same result as parallelogram addition of [i] the sum of the vertical components to [ii] the sum of the horizontal components.

This implies the component method for computing the components of the map or parallelogram sum of two arrows. Compute the horizontal and vertical components of the sum by adding the horizontal and vertical components, respectively, of the summands. The sum itself is then give by the map or parallelogram sum of its components.

Coordinates of Points and Arrows

Recall the rectangular and polar coordinates of points in a plane. These coordinates can be measured \rm with ruler and protractor. Measurement of these coordinates in a few examples is enough to convince us of their existence.

In the coordinate plane, each point is represented by a pair of rectangular, alias Cartesian coordinates $(a,b).$ Each point $(a,b)$ in the plane determines a vector with tail is at the origin $(0,0)$ and head at the point $(a,b).$ This vector represents the linear displacement from the origin to the point.

Polar coordinates $[r,\theta]$ define for each point a distance or length $r$ to the origin $0=(0,0)$, and an angle $\theta.$ The latter is a measure of the angle between the positive half of the horizontal axis, the line segment joining the origin to the point. Points in the plane can now be located using rectilinear displacements from the origin, or using counterclockwise angular displacements from the horizontal axis and a distance from the origin. Note the Pythagorean theorem implies \[r=\sqrt{|a|^2+|b|^2}\]

For convenience, we will also represent the point $(a,b)$ in the Cartesian coordinate plane by its polar coordinates $[r,\theta].$ That is, we write $[r,\theta]=(a,b)$ when both determine the same point. Here square brackets are reserved to indicate the polar coordinates of a point while round brackets indicate rectangular or Cartesian coordinates of the same point.

A point or arrow with tail at the origin be specified by giving rectangular or polar coordinates, or both.

Note the origin $(0,0)=[0,\theta]$ is a special case for polar coordinates. It is located by $r=0.$ The value of the angle $\theta$ has no effect and can be selected arbitrary. The arrow of zero length has no defined direction. Adding it to any arrow yields that arrow.

Coordinates and Components

Each point $(a,b)$ regarded as an arrow or vector, is the sum of two components: the horizontal component is determine by the point $(a,0)$ and the vertical component is determined by the point $(0,b).$

Coordinate Addition Method

The arithmetic sum of the rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2)$ is given by their arithmetic sum $(a_1+a_2,b_1+b_2).$ So we write \[(a_1,b_1)+(a_2,b_2) = (a_1+a_2,b_1+b_2) \] For example, $(2,3)+(5,7)=(2+5,3+7)=(7,10).$

Arrows equal in length to the components of two arrows being added are indicated in the following diagram. The previous conclusions drawn illustrated by this diagram.

The rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2)$ respectively identify two arrows from the origin $0=(0,0)$ of the plane. We will compute the sum of the horizontal and vertical components of these two arrows. The horizontal components are $(a_1,0)$ and $(a_2,0).$ Their sum is $(a_1+a_2,0).$ This gives the horizontal component of the parallelogram sum of the two arrows specified by the rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2).$

Similarly, the vertical components of the two arrows specified by $(a_1,b_1)$ and $(a_2,b_2)$ are $(0,b_1)$ and $(0,b_2).$ Their sum is $(0,b_1+b_2,).$ This gives the vertical component of the parallelogram sum of the two arrows determined by the rectangular coordinates $(a_1,b_1)$ and $(a_2,b_2).$ Finally, the parallelogram sum of the two arrows specified by $(a_1,b_1)$ and $(a_2,b_2)$ is the sum of its components $(a_1+a_2,0)$ and $(0,b_1+b_2,).$ The latter sum yields the arrow associated with the rectangular coordinates $(a_1+a_2,b_1+b_2).$ Thus the component method for addition of arrows agrees with the arithmetic method

\[(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2)\]

Thus the addition of arrows can be represented and done in terms of coordinates - The axes need not be orthogonal.

Real Multiples of Vectors

Multiplication By Positive Numbers or $-1$

Observe the addition of a point $(a,b)$ to itself yields $(a+a,b+b) =(2a,2b).$ The product of $(a,b)=[r,\theta]$ with a positive number $c$ is taken to be the point \[c\cdot (a,b) =(ca,cb)=[rc,\theta]\]

When $c=n$ is a whole number, multiplication by $c=n$ corresponds to adding the point $(a,b)$ to itself $n$ times. This can be shown using the principle of mathematical induction.

More generally, when $c=\frac nm$ is a rational number, multiplication by $c$ correspond to adding the point $(\frac1m a,\frac 1m b)$ to itself $n$ times. Here adding the point $(\frac 1m a,\frac 1mb)$ to itself $m$ times yields the original point $(a,b).$ Finally, multiplication by -1 is assumed to reverse the direction of an arrow and the linear displacement that the arrow represents. This reversal does not change the arrows length. Two reversals presumably yield the original direction. See the next diagram.

Note that integer multiples of the an arrow preserve the slope or direction or angle of the arrow. We further assume

$(-a,-b)=[r,\theta+180^\circ]$ if $(a,b)=[r,\theta].$

In general, the product of a point $(a,b)=[r,\theta]$ in the plane with a real number $c>0$ is $(ca,cb)=[cr,\theta]$ if $c>0$ and $(ca,cb)=(|c|r,\theta+180^\circ) = (-|c|a,-|c|b)$ if $c<0.$

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science