In the previous post, I dealt with vector addition. Here, I deal with subtraction and throw in a little work with currents.
I suppose I could just be glib and say that vector subtraction is just like vector addition, except you multiply the vector you want to subtract by -1 (minus one) and then just add that. This works for sure, BUT, it doesn’t give a person much intuition for vector subtraction, or why you would do it. So, I’ll teach a graphical method using an example, which may give you some intuition about how it works.
Here’s the setup, you’re paddling into a region that you know has a substantial current, but don’t know what it is. Your paddling speed is 3 miles per hour, and you paddle at a heading of 330o for one hour. After that one-hour, you use triangulation on landmarks to find your position and find that you’re at a northing of 3 miles and an easting of 1 mile. What’s the speed and direction of the current?
Figure 1 Setup of the problem to be solved.
Figure 1 shows a setup of the problem to be solved. If you paddle at a heading of 330o for one hour in the absence of any current you have the vector drawn in the figure. The fact that your position is not where you would have gotten purely from your heading, speed, and paddling time means that there is a current moving you. How do you figure that?
Let us imagine that we dropped a leaf in the water at the origin and we let it float for an hour. The leaf’s motion is completely due to the current in the water and we get the displacement of the leaf due to the current alone. The sum of the displacement of the leaf (current alone) and the displacement you get without current gives your real displacement. That is to say:
(real displacement vector) =
(leaf displacement vector) + (no current displacement vector)
But, what we do know is the (no current displacement vector) ie. the one drawn as the arrow, and also the (real displacement vector), So, in order to find the (leaf displacement vector), we need to revise the equation above to make:
(leaf displacement vector) =
(real displacement vector) - (no current displacement vector)
That is to say, we do a vector subtraction The way to do this is to place the vectors to be subtracted tail-to-tail (recall in vector addition we placed them tip-to-tail). Once we do that, we draw a vector between the tips of the two vectors, with the tail lying on the tip of the vector being subtracted and the top lying on the tip of the positive vector.
Figure 2 We’re subtracting the vector on the left from the vector on the right.
So, we draw a vector from the tip of the vector on the left to the tip of the vector on the right. The result is shown in figure 3.
Figure 3 The difference vector drawn in.
Next, we want to measure the length and direction of the difference vector, which represents the displacement of the leaf after an hour. To do this, we construct a set of axes at the tail of the difference vector and then measure the angle, and the length of it.
Figure 4 Finding the angle of the difference vector.
Figure 5 Finding the length of the distance vector.
From inspection using the protractor and ruler, I get the direction of the difference vector to be 80o and a distance of 2.5 miles. Now, since this is the difference that occurred over one hour, you just divide by one hour to get the current speed, which is 2.5 miles per hour. If we work in nautical miles, this would translate into 2.5 knots.
Some nautical terminology is in order here. When we talk about currents, we talk about their set and drift. Set is the direction of the current, and drift is the magnitude of the velocity or its speed, usually in knots. So, the current we just found has a set of 80o and a drift of 2.5 knots.
Comments on vector addition and subtraction
As I mentioned at the beginning, you can subtract a vector by multiplying the vector to be subtracted by -1 and adding it to the other vector to get a subtraction. The result is the same. I illustrate this below in Figure 6. Multiplying a vector by -1 is the same thing as swapping the tip (arrowhead) and the tail. So, what I’ve done in the figure is swap the arrowhead and tail of the vector to be subtracted and moved it over for the tail-to-tip addition rule (see previous installment) and then added it to the other vector. The resulting difference vector can be seen, and it also has an angle of 80o and length of 2.5 miles. This is just another way of doing the subtraction.
Figure 6 Subtraction by taking -1 times the original displacement vector and adding it.
On another note, I should mention that the process of vector addition is commutative. That means that you will get the same result if you reverse the order of vector addition. In other words
(vector a) + (vector b) = (vector b)+(vector a)
This is sometimes known as the “parallelogram rule” – as shown below in Figure 7.
Figure 7 Parallelogram scheme showing commutivity of vector addition.
The next post will deal with figuring out currents using vectors – again graphically.
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