Adding vectors

Adding vectors

In the previous installment, I wrote about the basic vector of a displacement.   In it, I used the example of a person who walked at 3 mph for an hour and 20 minutes at a heading of 70^o as an example, and asked the question, “what is the northing and easting at this position?”.   

Often, travels consist of multiple legs that can be approximated by straight lines.   Here are some examples:

1  1.  On land – detouring around an obstacle, like a cliff.   One might have to bushwhack first in one direction, and then turn to a new heading, and then keep track of the final position after a second or third leg of travel.

     2. At sea – sailboats often have to change direction to take advantage of favorable winds.   In the case of traverse sailing, a vessel may have to tack back and forth many times into the wind to make headway, but then in order to properly figure position with dead-reckoning, the navigator has to factor in all these tacks.

There are many other possibilities.   The point is that one has to add multiple legs of a trip and figure out the location.   Here is where vector addition comes into play.   There are multiple ways one can add vectors, but I will illustrate one graphical solution that is straightforward to carry out.   In terms of graphing tools, one useful item to have is a set of parallels.

Figure 1 A set of parallels.

Figure 1 shows a typical set of parallels.   These are and were a very common fixture among the navigator’s toolkit.   With it, you can draw a line parallel to another with high precision.    Lacking this, one has to rely on carefully using a ruler or a compass to establish parallel lines by “eyeballing”, which loses some precision, but one has to do the best one can with what one has. 

Let us assume that you’re going to just use a ruler to eyeball parallel lines.

Returning to our original example, you start at your base-camp, travel for one hour and twenty minutes at 3 miles per hour at a heading of seventy degrees.   Suppose now that you change direction and travel at a heading of 10^o for one hour, again at a speed of three miles per hour.   Where are you?   What is the straight-line distance to your based-camp?  What is your northing and easting in the new location?    Answering these questions is a matter of vector addition.  

That is to say:

First leg = 4 miles at 70^o

Second leg = 3 miles at 10^o

Where are you?

Figure 2 Vector addition.

The way you add two vectors is to place them tail-to-tip and then draw a line from the tail of the first vector to the tip (arrowhead) of the second vector.   This is shown in figure 2, and the resultant displacement after the two legs of the journey is indicated by the resulting displacement “Leg 1+Leg 2”.    Note that in general the resulting displacement is always smaller in magnitude than the sum of the magnitudes of the two displacements.   Only when the two legs are parallel do you get the two magnitudes adding up to be the magnitude of the resulting vector. 

So, how do you do this in practice?   First, there are math formulas one can use, but the idea is that you need to do this simply, so a graphic solution typically gives a good result.  Let’s then return to our first displacement from last time:

Figure 3 First leg.

We can extend the perpendiculars to create a new origin at the tip of the first leg (arrowhead). 

Figure 4 Vector extended perpendiculars.

Now, we take the protractor (or compass) and use the new lines to help us draw the 10^o line by making a tick mark at 10^o with respect to the new axes. 

Figure 5 Mark angle of vector to be added.

Next, use a ruler to draw the vector from the new origin for the distance required (3 inches = 3 miles) along the angle you just ticked off.

Figure 6 Draw the second vector.

Now, draw the vector by connecting the tail of the first vector to the tip of the second vector.

Figure 7 Draw sum by connecting tail of first to tip of second vector.

Now you have the sum of two vectors.   You can measure its properties.   First find its angle with a protractor.   In this case, I got 45^o.

Figure 8 Measure angle with a protractor.

Next, measure its length with the ruler (or compass edge) – I got 6 inches.

Next, find the components by dropping perpendiculars.

By inspection, I estimate the angle is 45^o, length is 6 inches = 6 miles, Northing is 4.2 miles and easting is 4.2 miles.   Precise numbers from calculator: angle is 44.7^o, length is 6.1 inches = 6.1 miles, easting is 4.2 miles, northing is 4.3 miles.   So, the precision is good to about 0.1 miles, consistent with the 0.2 precision we got in the last exercise.

Note that all the operations above can be done with a backpacker's compass.   Next, I'll discuss vector subtraction.   Also, as we go along, I'll have fewer and fewer of the explicit graphic instructions, as I'll assume that you begin to become conversant with these operations.

Exercise

This is a follow-on to the exercise in the previous piece on vectors - start out with this vector: You travel at a heading of 36 degrees for an hour and 10 minutes at a speed of 2 miles per hour.   Then you travel on a heading of 80 degrees for an hour at three miles an hour. What is your northing and easting after this time?