Introduction to displacement/vectors

In the previous post, I talked about dead reckoning, but now we can take a more abstract view of things and talk about how displacements can be represented as vectors.

Working with vectors

The word “vector” will strike terror into the heart of most mortals because it has a ton of possible definitions, many highly esoteric.   In this case, we’ll back way off and define it to be something useful in navigation: it can represent a displacement i.e. a distance and direction, or it can represent a velocity, which is a speed and the direction of motion.

Figure 1 A vector representing a displacement.

Figure 1 shows a representation of a displacement vector.   Vectors are graphically represented as an arrow with a ‘base’ and a ‘tip’.   The base-to-tip indicates directionality.   So, say you’ve traveled from a starting point and end up somewhere.   The base represents your starting point and the tip represents where you are after traveling some distance.  

Now, we come to the question of scale, which is inherent in any graphical representation of travels in navigation, typically on a map, but not necessarily.   You have to decide what distance on the ground or on the sea is represented graphically on your map, chart, or drawing.   Typical scales can range from large scale, like 1:1,000,000 (one to one million) or 1:24,000, which is small scale.   Maps and charts can even be as detailed as 1:10,000.   Just to be explicit, 1:10,000 means that one centimeter on the map equals 10,000 centimeters on the ground.

Sometimes in quoting scale, people may say that the map or chart has one inch equals a mile or one centimeter equals one kilometer.  In the latter case of one centimeter equals a kilometer, this would be a scale of 1:100,000.  In the case of one inch equals a mile, this corresponds to precisely a scale of 1:63,360.   A very common scale on charts and maps is 1:64,000, a holdover from the English system of units.   Using 1:64,000 is helpful because 64 has a large number of divisors.

Let’s return to Figure 1.   A long tradition of a system of coordinates is the easting-northing system.  An origin is established, somewhat arbitrarily, although there are conventions that we’ll discuss.   This is the easting=0, northing =0 point.

1.) Displacements to the east are positive eastings, displacements to the west are negative eastings.
2.) Displacements to the north are positive northings, displacements to the south are negative northings.

Now, it’s rare that one travels due east or due north on a journey, but rather some combination.   The vector in Figure 1 represents the displacement from the starting point to a destination with some combination of northing and easting.   The distance traveled is the magnitude of the vector, and the direction is indicated by the angle from the vertical or northing axis.   You would represent the distance by using the scale, and the angle typically is represented in degrees.

The convention in navigation is to take 0o (zero degrees) to be north, and then take the angle of the vector to go from 0o to 360o clockwise.   Part of this convention is rooted in history.   A sundial works by having the shadow of a stick or gnomon cast on a dial of some kind.   Since the motion of the sun in the sky is from east –to- west, the motion of the shadow is clockwise.   Mechanical clocks imitated the direction of the shadow passage.   The angle convention is a carry-over from this.  The choice of north as the zero of the angle likewise has historical roots.   In the northern hemisphere, at local noon, the sun casts a shadow due north.   Since this was an easy-to-define moment, it was logical to establish north as the zero of this angle, which we call the azimuth.

So, a displacement vector can be defined as a distance and an azimuth.  In addition, we could also specify a displacement vector by its components, which would be the amount of easting and northing.

Example

Let’s say you are in an unknown territory e.g. a flat woods or a foggy (calm) sea.   You start from your camp and travel for an hour and twenty minutes at a speed of three miles per hour at a heading of 70o.   Draw a representation of the vector of your displacement.

Step 1

Establish a scale.   You may have a map already, in which case you need to work with that scale.   You can make your own map.   Let’s think about the scale for a minute.   If you travel for an hour and twenty minutes at three miles an hour, you’ve gone four miles.   At three miles an hour, twenty minutes being one-third of an hour, is one hour every twenty minutes (yes, it should be obvious, but let’s just make sure we’re all on board).

A typical piece of paper is 8x10 inches.   I’d like to have the displacement be not so tiny that I can’t work with it, but not so large that it doesn’t fit, or doesn’t allow me the ability to put on more displacements.   So, I’ll choose one inch equals a mile, which is quite close to the standard 1:64,000 scale found on many topographic maps.

Figure 2 Setting a scale of one inch equals one mile.

So, get that piece of paper, get out that ruler and draw the length on your map of one inch equals one mile so you don’t forget.

Step 2

Establish an easting-northing coordinate system.   To start, choose an origin and draw a horizontal line for your easting.   Then, tick off each inch to establish a scale.   Obviously you could find some graph paper that a grid based and do the same thing, but let’s be as minimalist as possible here.

Figure 3 Setting the easting coordinates.

Next find a perpendicular from the origin that extends to the north – you can use a protractor to create a tick mark for the perpendicular.

Figure 4 Perpendicular tick mark.

Then use the tick mark to line up the ruler to create a northing scale.

Figure 5 Establish northing scale.

Label the scales.

Figure 6 Label scales.

Step 3

Now we actually draw the vector.   Remember that we’ve said that it’s 4 miles long at a heading of 70o.   Place the protractor back to the origin and draw a tick mark at 70o.  Here is an important point: people often get screwed up with protractors in drawing angles by not recognizing where the “zero” of the angle is located.   Make sure you have the “zero” placed properly, or recognize that you may need to subtract ninety degrees in order to get the correct angle.   Since this is an azimuth of 70o, and north is 0o, you’ll be closer to the Easting axis.  

Also note that in most math classes, the “x axis” would correspond to the Easting axis, and angles in math classes are typically labeled as going counterclockwise, started with zero from the positive x-axis.   This is not what we do in navigation – again zero is the Northing axis and angles extend clockwise for our purposes.

Figure 8 Use a protractor to draw a tick mark at seventy degrees.

Now lay the ruler with the 0 at the origin, extend through the tick mark and draw a line 4 inches long.   This becomes your vector.

With your vector, you might want to know your northing and easting.  Northing and easting were (and are) common terms for nautical and land navigation and were used quite frequently.   In the modern era, there is a grid system used to identify coordinates anywhere on Earth called UTM for Universal Transverse Mercator.   It covers the Earth with a set of grids that have a well specified origin and coordinates are given as positive numbers in positive distances of eastings and northings from the origin.  UTM coordinates can be found on the sides of topographic maps and nautical charts.
A close cousin of UTM is the Military Grid Reference System, or MRGS, which is used by NATO countries to identify coordinates.

Figure 8 Use the ruler to draw the vector by extending the seventy degree line four inches.

Drawing with a compass

You can do all the above operations with a magnetic compass.   Using the base and housing, which rotate, you can make angles, and the baseplate is typically marked with scales – inches, centimeters, and also miles for common scales, like 1:24,000 and 1:64,000.    I illustrate the operations above with a protractor and ruler, as they tend to give more precise results.

Let’s take the placement of the 70o tick-mark as an example.   Line up the orienting lines on the compass housing parallel with north.   Rotate the baseplate until the back-end of the direction-of-travel arrow lines up with 70o, and move the edge of the baseplate until it intersects the origin.   Then you can place the tick-mark.

In effect, you can do everything with a compass that you can with the ruler and protractor.   I won’t redo all the above steps with the compass, as I hope it’s self evident.

Figure 9 Using a compass as a protractor.   Here I illustrate how to place the seventy degree tick mark with the compass (see text). 



Finding components (or projections)

Having traveled for one hour and twenty minutes on a heading of seventy degrees, you might want to specify your location to a friend over a radio in terms of easting and northing.   How would you figure this out?   Here is where we have a couple of choices.  In one case, we could use sine and cosine functions from trigonometry, but these require a calculator.   The other way to do this is to make a graphical solution.  Since there is an inherent uncertainty in how fast your walk and maintaining your heading, the graphical solution is good enough for our purposes.

The main idea in the graphical solutions is to draw lines perpendicular to the easting and northing lines that intersect the tip of the arrow.   These give the vector components.  Figure 10 shows how to do this for the easting component.  Place the ruler on the tip of the arrow and keep the ruler parallel to the northing axis and draw a dotted line down to the easting axis.

Figure 10 Dropping a perpendicular to the easting axis from the displacement vector using a ruler. 

Note that you won’t be able to get the ruler perfectly aligned with the northing axis, but using the tick marks and a bit of effort you can get decent results.   You can likewise do the same thing by running a perpendicular to the northing axis using a ruler parallel to the easting axis.

Figure 11 shows the results of making both perpendiculars to extract the northing and easting component.   Now, to state the results, there is the tricky business of visually estimating the size of the components.   People can typically subdivide an interval into about 10 equal parts, and then lose precision at this point.   What do you think the components are?   For me, I’d say that the easting is 3.8 miles and the northing is 1.2 miles.

If I go to my calculator, I find that the easting is 3.8 miles and the northing is more like 1.4 miles.   The difference between 1.2 and 1.4 is “in the noise” as they say.   That is to say, sloppiness in drawing and visual estimates gives me an uncertainty of about 0.2 miles.   It’s good to keep this in mind when making graphic solutions, or any estimates of numerical quantities.   Note, too, that I’m only using two significant figures.   Higher precision is clearly not warranted.

Figure 11 Easting and northing components of the displacement vector.

Exercise

You travel at a heading of 36o for an hour and 10 minutes at a speed of 2 miles per hour.   What is your northing and easting after this time?

Next: Adding vectors