# Vector Thinking #1

Last time I talked about opposing pairs: Yin and Yang, light and dark, north and south. I mentioned that some pairs are true opposites of each other (for example, north and south), whereas other pairs are actually a thing and the lack of that thing (for example, light and dark). Such pairs are only opposites in the sense that an empty cup is the opposite of a full cup.

However in both cases, the opposites stand for opposing ideas; two poles of polarity, and it is polarization that I address today. Specifically I want to discuss a way of thinking that helps avoid it.

It’s easy to divide the world into sides. Many sayings begin with, “There’s two kinds of…” It seems easier to break things down into opposing points of view than to consider a variety of views. It seems easier to compare features between two things than twenty. Our court system has two sides and so does our political system (despite many attempts to create a viable third party).

Many of our social issues are also seen as an opposition between two poles. You either believe this or you believe that. If you find points on both sides of a question, it can feel like you fall in the middle with no opinion at all. In fact, you may feel strongly about both sides!

There is a way to think about this that doesn’t make you feel like you fall into the zero zone in the middle. Basically you have to get off the one-dimensional “number line” and think two-dimensionally. I call this Vector Thinking.  I’ll explain it in a moment, but first a closer look at the problem.

### Push/Pull Thinking

figure 1

Tricky social issues are sometimes seen as a zero sum situation. There is a perception of a fixed resource, the opinion, for which there is a competition. To the extent one side wins, the other side loses.

A way to visualize this is to see an issue on a traditional number line. You may remember number lines from school. There is a horizontal line with zero in the center, negative numbers extending to the left and positive numbers extending to the right. Figure 1 shows such a line.

figure 2

Figure 2 shows a number line representation of the difficult social issue: Green tea or Brown tea. Arbitrarily, Green is on the left, and Brown is on the right. The more you choose Green, the less you choose Brown. This works fine if you’re a dedicated fan of one to the exclusion of the other.

The middle might, at first, seem neutral territory; the place of no opinion either way. But what if you like Brown tea just as much as Green? What then? On the number line, you would seem to have zero opinion.

The problem is that your vote is represented by only one number. Somehow you have to sum up your feelings in a way that allows you to “stick a pin” on the number line. In the example, the vote is cast mainly in favor of Green tea. But the vote isn’t as far to the left as it could be.  Does this mean the voter doesn’t like green tea all that much, or does the allure of Brown tea pull them right? There’s no way to know!

And what if the vote were cast at the halfway mark, at zero?  Does that mean the voter doesn’t like tea at all, or that the voter likes them equally? And if the voter does like both types of tea, is the liking strong, medium or weak?

With number line thinking, there’s no good place to stand if you find merit in both sides. Moving to the middle instinctively feels like moving towards the other side. Standing in the middle feels like nothing at all. Zero.

We need a better metaphor.

### Two-Dimensional Thinking

figure 3

If you remember the number line, perhaps you remember the X-Y graph.

Figure 3 shows a typical X-Y graph. Rather than a single number line, X-Y graphs have two number lines, one set at 90 degrees to the other. The X axis runs horizontally along the bottom, from left to right. The Y axis runs vertically along the left side, from bottom to top.

A number line is one-dimensional; the pin you stick into a number line specifies one number. X-Y graphs are two-dimensional. When you stick a pin into an X-Y graph, you specify two numbers! In fact, the terms, one-dimensional, two-dimensional and three-dimensional refer to the number of numbers required to stick a pin into any given point.

figure 4

Figure 4 shows a new way to look at the Green/Brown issue. Now, because you specify two numbers when you stick the pin, you can vote for Green and Brown separately. Instead of having to pick one spot on a number line, you can cast two votes with a single pick.

In the example, we can see that the voter is just past the halfway point on liking Green tea and not to the halfway point on liking Brown tea. We can see the voter likes tea okay (but not hugely) and prefers Green to Brown.

Looking at things this way provides four basic zones for a vote. The upper-left zone indicates a strong liking for Green tea with little interest in Brown. The lower-right zone indicates the opposite: a love of Brown tea with low interest in Green. If someone loves both types a lot, their vote falls in the upper-right, and if neither tea holds much interest the vote falls in the lower-left.

Votes along the axes indicate zero interest in the opposing axis. For example, if you don’t like Brown tea at all, but do like Green, your vote would fall somewhere on the vertical axis on the left.  How high it would be depends on how much you like Green tea. Likewise a preference for Brown with no interest in Green would fall on the horizontal line along the bottom.

If you don’t like tea at all, your vote would fall at the zero-zero point on the lower-left.

### Vector Thinking

figure 5

Two-dimensional thinking helps think about issues that have two sides, but the idea can be extended to more than just two axes. It’s difficult to represent more than two dimensions on a two-dimensional page, but consider figure 5. This is a “flattened” image of a three-dimensional X-Y-Z graph.

I’ll continue this next time (I need to create better graphics first). For now I want to explain the term, “Vector Thinking.”

Technically speaking, a vector is a bunch of numbers that together provide a useful piece of information. For example, if you combine the GPS coordinates of your current location along with numbers reflecting the direction you’re moving and how fast you’re moving, that’s a vector that both locates you and describes where you’re headed and how fast.

Think of a vector as an arrow pointing away from you. The base of the arrow is your location, and the length represents your speed. The longer the arrow, the faster you’re going.  Your direction is where the arrow points. You could draw such an arrow on a map, and that arrow would encompass all three aspects of your course: your current location, your direction and your speed.

Here’s what’s important about vectors for now: when you look at an X-Y graph (or an X-Y-Z graph), the “X” and “Y” (and “Z”) axes are vectors. Commonly both start from “0” and point away in perpendicular directions (usually upwards and rightwards in the case of an X-Y). The axis vectors are often “infinite” in that they could extend (in theory) forever.  In some graphs, there is a maximum.

But any specific point on the graph is represented by specific pairs of vectors along the axes. Where those vectors end marks the point along the axis for the value along that axis.  For example, in our Green/Brown graph, the vertical vector has a value of 4.5 and the horizontal vector has a value of 3.0.

On the three-dimensional graph, there are three vectors. The X-vector has a value of 2; the Y-vector has a value of 3; the Z-vector has a value of 5. The specific point of interest on the graph is a “vote” of 2, 3 & 5.

I’ll let you chew on this until next time!