One of the things that mariners regularly do is measuring their surroundings. We measure the air’s temperature, the water’s depth below our keel, the boat’s speed, and even the cost of a new boat.
In most cases, the result of these measurements is a single entity, which describes the amount or magnitude of something at the point and time of measurement. We are all familiar with these scalar quantities: temperature (37° F), depth (12.5 feet), speed (20 feet per second), boat price ($1,000/foot, if one is lucky). Scalar measures are very common, and our brains are well adapted to understand their significance.
Many people seem less comfortable with the vector. This is a measurement consisting of two numbers: a magnitude and a direction. Boat velocity (five knots on a course of 354° T), and current set and drift (2.5 knots to the northeast) are both vector measurements, because they give the direction in which something acts or moves, and the magnitude of that motion.
A navigator’s day is literally filled with vector measurements, and the difficulty some sailors have doing proper navigation testifies to our brain’s lack of an intuitive grasp for how vectors work. We’ll address that lack by introducing vectors as tools for understanding relative motion problems. The mathematically challenged are particularly encouraged to read onone of the more attractive features of vectors is that in the vast majority of cases, they can be manipulated graphically, using only dividers and a straight edge. No calculators, tables, or black boxes are required.
A vector requires both a direction and a magnitude, but it is important to realize that a vector is really one measurement having two parts. Representing vectors as arrows on a two-dimensional surface, such as a chart or workbook, demonstrates this well. In figure 1, vector A is shown by one arrow, but that arrow contains two pieces of information: the direction and magnitude. To fully understand vectors, one must consider them in this manner.
A vector is a straight line, and there are two diametrically opposed directions given by a line. By convention, a vector is drawn with its origin, or tail, at the point of measurement, and its head (shown by the arrow) in the true direction of the vector. Navigators customarily use azimuth to indicate direction; unfortunately, mathematicians do not, and it was they who developed vector mathematics. No matter; we will recall that one of the great early American mathematicians was Nathaniel Bowditch. Therefore, we’ll follow the navigator’s convention of using azimuth for direction. In all of the figures we’ll assume that north is at the top of the page, and measure azimuth clockwise from north.
All vectors must have a length (even if that length is zero), and to measure the length we need a scale. In this article, we will assume a scale of one inch equals five knots or five nautical miles, depending upon the problem. Do not be confused by our use of a scalar measurement (a scale) to measure an aspect of a vector (its length). We did that in the previous paragraph, when we used the scalar “azimuth” to measure the vector’s direction. Using dividers or a good eye, one can measure or estimate the magnitude of a vector using the scale. The scale can be used with a vector of any direction, just as one uses the latitude scale of a chart to measure the distance of any track, regardless of its azimuth. To measure the length of the vector, move and rotate the dividers to the scale as required, and read the magnitude from the scale. Vector A in figure 1 is 7.5 knots to 065° T, while vector B is 3 knots to 120° T.
Variety of vectors
Vectors come in a variety of types, each suited to a particular task. In this article, we will consider only “free” vectors. Free vectors can be moved (but not rotated or stretched) at will across your chart or workbook without changing the vector. This is essential for everything we will do here, and it is really no different from saying that 78° F in Portland is the same as 78° F in Miami (although it may feel different to the locals). Thus, in figure 1, vector A is the same as vector A’. Let us make a bold mathematical statement about any vectors that have the same direction and magnitude: They are equal, or as in figure 1, A = A’. Not all vectors have this freedom. For example, the net effect of the wind upon a sail acts at a specific point on the sail, in a specific direction, with a specific force. This force is thus a vector, but the boat would respond differently if the force were moved from the sail to (for example) the hull. We will studiously avoid these “fixed” vectors.
The advent of calculators notwithstanding, most people are comfortable adding, subtracting, or multiplying scalars. Performing these tasks with vectors is also straightforward, because for the purposes we will discuss here, addition and subtraction of vectors is identical to the addition and subtraction of scalars. We have the added benefit that vectors can easily be manipulated graphically as well.
Graphically, vector addition is straightforward. For example, to add vectors A and B , place one vector (it doesn’t matter which) anywhere on your worksheet, maintaining, of course, its direction and magnitude. Now, place the second vector’s tail at the head of the first vector, again maintaining the direction and magnitude of the second vector. This process creates a new vector, C, from the tail of A to the head of B figure 1. This procedure can be written in symbols: C = A + B. As in scalar addition, the order of vectors in the addition doesn’t matter: C = A + B = B + A. Also, just as in scalar addition, one can add as many vectors as needed, in any order, by placing the tail of the next vector at the head of the previous one. The vector resulting from all this addition is the resultant, and it represents the net effect (or result) of all of the summed vectors. Note how the graphic summation of A and B to form C in figure 1 expresses the algebraic equation C = A + B. Vector addition and subtraction can be done symbolically in equation form or graphically; both methods are different ways of performing the same task.
Here are two examples, using displacements rather than velocities, to further illustrate this process. Remember that north is toward the top of the page, and that we are using a scale of one inch = five nm. One makes five miles along a course of 070° T and then six miles along 030° T. What are the net distance and course made good? Figure 2 shows the mechanics: the answer is 10 miles to 050° T. What about if we reversed the order of addition: 6 miles along 030° T followed by five miles along a course of 070° T. As one can see by plotting it, the resultant is the same, because the order in which vectors are added is irrelevant. Remember: To add vectors, attach them head to tail, and draw the resultant from the first vector’s tail to the last vector’s head.
Vector subtraction is the opposite of addition, but subtraction of vectors can be confusing. There are a number of methods for doing it, and we will use the one I feel works best for navigation (and easy presentation). Consider the difference A – B = D. To subtract from vector A the vector B, place the heads of the two vectors together. The resultant D is the vector from the tail of A to the tail of B. Note the order: heads together, then from the tail of the first vector to tail of the second.
The only difference between vector subtraction and vector addition is the location of the second vector. Compare A – B = D to A + B = C in figure 1. To subtract, place the head of the second vector at the head of the first vector; when adding, place the tail of the second vector at the head of the first vector. Always draw the resultant from the tail of the first vector to the unoccupied end of the second vector. Other methods are possible, but if one is using a plotting sheet for a workbook, this process allows the use of the compass rose for both vectors, which shortens the process by a small amount; this method also seems easier to remember.
So, here is an example of vector subtraction: Over a two-hour period we made 4.5 miles towards 220° T; in the first hour we made 2.5 miles towards 260° T. What was distance and course made good in the second hour? The first hour’s travel added to the second hour’s travel equals the total distance. Therefore, we face a subtraction problem: from the total distance made good we must subtract the first hour’s progress. Figure 3 shows this case graphically; the answer is 3.5 miles to 190° T. Remember: to subtract, place two vectors head to head, and draw the resultant from the first vector to the second.
Multiplication
Multiplication of vectors is also straightforward. Multiplying a vector by a scalar (that is, “regular”) number changes only the length of the vector. Vectors multiplied by negative numbers are not only stretched (or shrunk), but they also reverse direction.
Figure 4 shows how this is done. We start with vector E, multiply it by two to get F, and by -1/2 to get G. A single example will illustrate the process. One has made five miles to 150° T in the last 60 minutes; how far, and in what direction, will one go in the next 30 minutes? Thirty minutes is one-half of 60 minutes, so the question requires multiplying the vector five to 150° T by 1/2. Multiplication doesn’t change a vector’s direction, so the vector is 2.5 miles to 150° T.
It should be fairly obvious from the above examples that dead reckoning is an exercise in vector algebra. Remember, to multiply a vector by a scalar, stretch or shrink the length of the vector by the scalar; if the scalar is negative, reverse the direction of the vector as well.
One of the wonderful benefits of using vectors (which, recall, are objects composed of two parts) is that they can be added, subtracted and multiplied just like scalars. Perhaps a group of vectors has the following relationship:
A + B – C = D – 2E – F/2
One can re-arrange this equation like any other equation; so one could solve the equation above to determine the vector D = A + B – C + 2E + F/2. One may not know what vectors A, B, C, D, E or F are, but one does know that as long as the first equation is correct, so too is any equation one can derive from it. In a sense, vectors allow one to manipulate symbolically items such as velocities or forces that aren’t scalar; items that one might not think could be handled easily.
With this compact but useful set of vector tools, we can develop a vector approach to relative motion. Relative motion is the apparent (or relative) movement of one boat as viewed from another. In essence, in determining relative motion one can assume that one’s own boat is stationary, and then determine all velocities relative to that fixed position. Boat velocity is a vector. A navigator is concerned with both the speed and course of his or her vessel. Because of this, we can use vector mathematics to determine the motion between two (or more) vessels moving in relation to each other, even if we happen to be on one of those vessels. In effect, understanding the relative motion between vessels can be reduced to vector addition, subtraction, and multiplication.
Rather than memorize the vector mathematics of relative motion, let us derive it.
We are sailing due north at five knots (our true velocity); directly ahead is another vessel sailing due south at six knots (that vessel’s true velocity). What is the relative motion of the other vessel? Common sense tells us that the other vessel’s relative velocity, as viewed from our cockpit is 11 knots directly southward. Now, try to determine the relative velocity with vectors (figure 5), remembering to keep the directions and magnitudes correct. Our velocity is shown by a vector five units long (choose a convenient scale) pointing due north, the other vessel’s velocity by a vector six units long pointing due south, and the resultant by a vector 11 units long pointing due south. The only way to arrange the three vectors in a correct fashion is by subtracting from the other vessel’s velocity vector our own ship’s velocity vector. Stated in words: a vessel’s relative velocity = vessel’s true velocity ⁄ own true velocity.
General vector equation
This result is a completely general vector equation. It works regardless of the speed or direction of one’s ship or the speed and direction of the other vessel. The equation above appears ungainly, but two manipulations will help. First, note that the relative velocity of our boat is missing from the equation above. Why? From the point of view of one’s boat, the relative velocity is always zero. With that in mind, recall that any equation can be manipulated algebraically, and one of the easiest manipulations in all of algebra is to add zero to one side of an equation. Since we know that our own relative velocity is zero, let’s add our own relative velocity to the equation. Also, group the other vessel’s velocities on one side of the equation and one’s own velocities on the other. With these changes, and omitting the word “velocity” from all of the terms:
Vessel’s relative – vessel’s true = own relative ⁄ own true
All these words obscure the mathematics, so let’s use symbols to indicate the vectors: VR for the other vessel’s relative velocity, VT for the other vessel’s true velocity, and OR and OT for own ship’s relative and true velocities. In symbols, then:
VR – VT = OR – OT
One can see why the symbols help. In vector-speak, the difference between another vessel’s relative and true velocities is equal to the difference between your own relative and true velocities. (Remember that one’s own relative velocity, OR, is always zero.) Let’s call this the “relative motion equation.” It is all one needs to know to handle any relative motion problem, as long as one feels comfortable moving terms from one side of the equation to another. One will generally know three of the terms in the relative motion equation: one’s own relative velocity (OR = 0), one’s own true velocity (OT), and either the other vessel’s true (VT) or relative (VR) velocity. One will be interested in finding the one not known. To do so, simply manipulate the relative motion equation as needed, plot the vectors, and find the resultant.
For example (figure 5), one is tracking a vessel on radar. The vessel’s relative velocity is eight knots to 310° T. One’s own velocity is four knots to 180° T. What is the vessel’s true course and speed? We know OR, OT and VR; we must find VT. First, we solve the relative motion equation for VT: VT = VR – OR + OT. Then plot the vectors. The resultant is 6.1 knots to 280° T, which gives the true speed and course of the other vessel.
In figure 5, one may have noticed, and reacted with alarm to, the ominous meeting of the OT and VT vectors at a point. Perhaps you fear that collision is imminent? The vectors do not indicate a collision is coming. Remember that the vectors we use are “free”; they can be moved at will across the paper. One could, if one wanted to, move any of the vectors anywhere else on the worksheet. So, the head-to-head meeting of the two true velocity vectors is not necessarily indicative of a collision. The previous sentence is true when read backwards: An imminent collision is not apparent when working with relative velocity vectors. These vectors represent the immediate relative velocity of another vessel. To determine the probability of a collision, one must determine the bearing and distance (a vector!) of the vessel. Do not get the two things confused: Velocities are free vectors, but bearings are “fixed” vectors. As implied by the truism “beware the constant bearing,” a collision becomes evident only after establishing the long-term relative bearing vector of another vessel. Relative velocity vectors will not confirm or deny the possibility of a collision; only plotting of bearings shows that. Remember then that understanding relative velocities will notby itselfprotect one from collisions.
Another example. We know from a VHF “securite” call that a vessel nearby is traveling six knots to 124°. We are traveling 5.5 knots to 060°. What will the vessel’s relative velocity be? Again, solving the relative motion equation for VR, we have VR = VT + OR – OT, and the resultant is 6.2 knots to 178°. Relative to us, the other vessel will be traveling nearly due south at six knots.
Vectors relative to each other
The relative motion equation is even more general than I first let on. In complicated traffic, it can be helpful to know how other vessels (all moving in different directions and at different speeds, after all) are viewing the situation. What, for example, is the relative velocity of that ship over to port as viewed from that boat over to starboard? As a first step, consider what our relative velocity is from another vessel. Returning to our original relative motion example, that boat north of us barreling along at a relative velocity of 11 knots due south sees a picture similar to ours. Relative to their boat, we are barreling along at a relative velocity of 11 knots due north. Our relative velocity as viewed from another vessel is simply the negative of that ship’s relative velocity as seen from our boat.
More generally, we can easily find the velocity of one ship relative to anotherwithout knowing either of the other’s true velocities. The nomenclature can get a bit bewildering here, so let us designate the two vessels as B and C; call our own vessel A. We will designate the various velocities as we did above, but add subscripts to indicate the boat to which the velocity belongs. For example, VR is B’s relative velocity. To determine the relative velocity of B as seen from C, subtract from the relative velocity of B as seen from A the relative velocity of C as seen from A (as I said, the nomenclature is a bit confusing). In symbols: (from A) – VR (from A)
This makes sense, in a way. Relative velocities assume that the observer is fixed that everything, including the ocean, is moving relative to the observer. In this case, our goal is to view the situation from the perspective of C, by shifting our frame of reference from our boat to the cockpit of C. So, to find the relative velocity of B from C, subtract away the relative velocity of C from the relative velocity of B. This has the effect of making the relative velocity of C zero, which is what we need to do. The equation above is completely general, and will work with any set of relative velocities.
Of all the cases we have considered, this one is most in need of an example, shown in figure 6. You have established the relative velocities of two vessels on your radar: vessel B has a relative velocity of 6.5 knots to 297° T; vessel C has a relative velocity of 7.5 knots to 169° T. What is the relative velocity of B as seen from C? The sketch is shown in figure 6; the answer is 12.5 knots to 325° T. If one tries this and comes up with an answer of 12.5 knots to 145° T, then the vectors were subtracted in the wrong order. If one comes up with something towards 224° T, then the vectors were added instead of subtracted. Note that we did not determine or even consider the true velocity of either vessel in this example, although one can do that too.
As with many aspects of navigation, proficiency in vectors requires practice. Perhaps the best sources of additional material are review guides for college algebra or calculus, which are available in paperback at most bookstores. I keep small versions of figures 1 and 5 in my navigation notebook, just in case my mind fails me and I forget the mechanics of vector algebra.
Larry McKenna teaches geology at the University of Kansas when he is not navigating.
