Put on your arithmetic caps, dear readers. Also your math mittens, geometry galoshes and cosine coats. Today we’re venturing after numeric prey that lurks down among the lines and angles.
There’s no danger, at least not to life or limb, but I can’t promise some ideas won’t take root in your brain. There’s a very real danger of learning something when you venture into dark territory such as this. Even the strongest sometimes succumb, so hang on to your hats (and galoshes and mittens and coats and brains).
Today we’re going after vectors and scalars (and some other game)!
Actually, vectors and scalars are pretty easy. It’s applying them in different contexts where things can get complicated. Simply put, a scalar is one number, a vector is a bunch of numbers about one thing. Unfortunately, putting it that simply doesn’t tell us much.
So let’s talk about speed (which is a scalar) and velocity (which is a vector).
For most, speed and velocity likely mean the same thing: how fast something is going.
If someone asks our speed, we might reply, “I’m going 40 miles per hour.”
If they asked for our velocity, we might look at them funny, but would probably still give the same answer.
Technically speaking (and we are speaking technically here), 40 MPH is an incomplete answer to, “What is your velocity?” A complete answer requires specifying speed and direction. Velocity is a vector consisting of two parts: speed and direction.
Speed — because it’s a scalar — is just a single number. That number has units — miles per hour in this case. Those units consist of two qualities: miles and hours. The name, miles per hour, means some number of miles traveled in one hour (the single hour is assumed).
So even a simple scalar like speed can have a little complexity under the surface. Compare this to simpler scalar value, weight.
If something weighs 50 pounds, then “50” is just a count of how many. Here there is no sense of “pounds per something” — it’s just pounds.
We would call speed a rate, whereas weight is a magnitude. The word, “per,” is the give-away word for a rate: miles per hour, pounds per square inch, typos per page.
Note that the number of miles traveled, the number of pounds pressing, and the count of typos, are all magnitude scalars. A magnitude per some other (single) unit creates a rate.
Now consider velocity, which is defined to have both (the scalar) speed plus another component: direction. Having more than one component makes velocity a vector.
It turns out that direction is a vector on its own. It requires more than one number to specify a direction. How many depends on how many dimensions your space has.
When Captain Kircardwayer, flying through three-dimensional space on his spaceship, the Voyerprise, calls for a heading (a new direction) of “90 mark 45” this specifies two numbers — 90 and 45 (the word “mark” merely separates them).
One way to describe a direction uses angles from some reference directions. One common technique uses azimuth and elevation, both of which are angles relative to our current position. Think of the azimuth as rotation, turning to our left or right. And elevation is how much we look up or down.
There are two basic approaches. The angle can be from 0 up to (but not including) 360 — a complete circle. Or it can go plus and minus, usually from 0 to ±180. There are pros and cons to both.
As it turns out — at least in three-dimensional space — it always takes two numbers to specify a direction from a given point. It always takes one less number to point from somewhere than it takes to be somewhere. In five-dimensional space, for example, it takes four (five minus one).
The reason is that direction lacks a notion of distance. We might point at something obvious, but the pointing itself does not imply any distance. This lack of a magnitude accounts for the minus one aspect of a direction (vector) compared to a location (vector).
In my Dimensional Coordinates post, I described how a location in three-dimensional space requires three numbers to specify. Now I can mention that those numbers comprise a location vector.
A key point of that other post was that dimensions and coordinates are related.
A coordinate is a point along some dimension. So, to specify a point in space, we always need as many coordinates as we have dimensions.
They can be X-Y-Z, or a pair of angles (such as longitude and latitude) plus a distance. We call these Cartesian or polar coordinate systems respectively. There are other ways to specify location and direction, but they all tend to require N and N-1 numbers (respectively) where N is the number of dimensions.
Let’s get back to our two-number direction vector. Using angles for both numbers is common. We can also project those two angles onto an imaginary surface surrounding us and think of them as “X-Y” coordinates on that surface. (Regardless, it’s still just two numbers.)
This is exactly what’s going on with longitude and latitude. We often think of them as being like “X-Y” coordinates, but both are actually angles relative to the center of the Earth.
Notice that these use the plus-minus angle forms. Latitude is plus and minus elevation north and south. Longitude is plus and minus angle west and east.
Obviously, these systems require reference directions to call “zero” — the angles are measured relative to the references.
On a flat surface (which has only two dimensions) it takes just one number (two minus one) to specify a direction away from some point. This is because we can only move away from a point along some single angle.
If the reference direction is straight up, we can measure any direction away from a point on a flat surface as an angle relative to straight up being 0 (zero) degrees (we’ll also assume the angle goes clockwise).
If we now jump up to four dimensions, a direction requires three coordinates. If time is our fourth dimension, pointing requires a time coordinate. “My keys were over there yesterday.”
And note again that pointing towards where the keys were does require three numbers (four minus one). Specifying the keys actual location requires all four coordinates. “My keys were in that place (X,Y,Z or whatever) yesterday.”
An important aspect of these coordinate systems is that they assume a center point — the origin of the coordinate space. It’s the point where all the coordinate values are zero.
This creates an interesting situation: a location vector — a point in space — has an associated magnitude vector that describes the location’s distance and direction from zero.
That vector comes from imagining a line from the zero-coordinate origin to the X-Y-Z point in space. We’re actually using six numbers, but we’re assuming half of them (3) are zero.
Really, this is just a way of looking at the three location coordinates. What’s added is that it’s easy to calculate the length of the line. That gives us a single number (a magnitude scalar!) that represents the distance of the point from the origin.
More to the point, vectors frequently have a sense of “arrow-ness” to them. We’ve seen that direction vectors literally point somewhere (from a given point). Now we see that location vectors also imply a direction.
It turns out that vectors often involve pointing. Consider the difference between air temperature — a magnitude scalar — and wind direction — a direction vector.
At whatever resolution we want to consider, each point in the atmosphere has a temperature and a wind direction.
The temperature is just a number; it has no direction. But wind direction does have a direction and requires two numbers to describe.
Wind speed adds a third number, and wind speed plus wind direction gives us, ta da, wind velocity!
So if anyone asks you what your velocity is, you can correctly reply that you’re going 40 miles per hour on a plus-three-degree slope with a +90-degree heading (that is, going east and slightly up-hill). “40 mark +3 mark +90!” you can exclaim.
Stay three-dimensional, my friends!
 This is different from scalars with multiple units (such as miles and hours). Vectors have multiple numbers. Scalars always have only one number; the other units are assumed: 40 miles per (one) hour; 50 pounds per (one) square inch; 32 feet per (one) second squared.
 The first approach suffers from a duplication of directions. For every direction with azimuth [180–360) and elevation [0–90) or (270–360) — that is, the entire left hemisphere — there is an identical direction with azimuth [0–180) and elevation (180–270).
The second approach works better, although it’s lop-sided. Azimuth is ±180 while elevation is ±90. There is also that -180 and +180 azimuth are identical. Verbally, the need to specify “plus” or “minus” is cumbersome.
Note an oddity of polar coordinates: degenerate cases where value of one (or more) coordinate(s) doesn’t matter.
When the distance of a polar system is zero (that is, in the center), both the longitude and latitude don’t matter. Likewise, at the poles (90° latitude), the longitude doesn’t matter.
(These degenerate conditions can result in gimbal lock.)
 There is an abstract way to reduce any multi-dimensional space to a one-dimensional coordinate system using space-filling curves, but [A] abstract and [B] beyond the scope of this already long post.
 Unless the time coordinate is the magnitude being left out. If we say, “My keys were there before,” we’re pointing backwards in time but not specifying a time coordinate.
 Dimensions can be abstract rather than physical. Time can be considered a dimension, and 4D+ “physical” spaces cannot be visualized. This is a place where math goes beyond the world as we see it.
 Taking it to the next level, consider a (straight) path through (3D) space. It has a start point and an end point. We could specify them as X-Y-Z locations. That means a 3D path requires six numbers. If we try having an X-Y-Z start point plus a direction (two more numbers), we still need a path length, and that brings us to six numbers.
This makes sense. In geometry, a line is defined by two points. In 3D space, a point requires three numbers to specify (because we’re in 3D space). So a line requires six numbers.
This holds true in 2D. On a flat surface, a point takes two numbers to specify (e.g. X-Y). A line is still defined by two points, so a 2D path requires four numbers. Thinking of it as a start point (X-Y) plus a direction angle plus a length still takes four numbers.
It even works in 1D! On a line, a point only requires a single number to specify. Two points on that line specify a path on the line. (The logic also extends into four (and more) dimensions, but we’re unable to picture those in our mind!)
The bottom line: A (straight) path requires twice as many coordinates as dimensions. A line can be objectively specified by naming its end points, or subjectively by naming the start point, direction, and distance.
 Assuming that actually is the way you’re going.