Today we’re talking about vectors. Why? Well, I can absolutely tell you that it’s not because they’re uproariously funny. I tried my best to find a joke about vectors for you and I came up almost empty-handed. Sure, there’s this one:
Question: What do you get when you cross a mosquito with a mountain climber?
Answer: Nothing. You can’t cross a vector and a scalar.
But I’m pretty sure that’s only funny to 1% of the math fans out there—although for that 1% it is rather funny … am I right? The truth is that vectors just aren’t the great comedians of the math world. And that’s okay! Because it turns out that vectors are more than useful enough to make up for their lack of funny.
How exactly are they useful? And what are they in the first place? Those are exactly the questions I’ll be tackling.
Why Are Vectors Useful?
Vectors are incredibly useful and have tons of real world uses in their own right.
The eagle-eared and elephantinely-memoried among you will remember that I recently discussed complex numbers. As you’ll recall, those are numbers made by adding a real number and an imaginary number together. At the end of our discussion of these complex numbers and the complex plane within which they live out their days and nights, I mentioned that complex numbers are not just a cool but perhaps crazy and abstract mathematical idea, but that they actually have very important real world applications. And I mentioned that we’d be talking about some of these applications in the coming weeks.
And we absolutely will be doing that soon! But before we dive deeper into those real world uses of complex numbers, it will be useful if we first talk about the aforementioned mathematical concept called a vector. Which is … ?
What Are Vectors?
When you hear what a vector is, you might be inclined to think that it’s too simple a concept to be of much use. But you would be wrong. Here’s the gist: a vector is nothing more than an arrow that has a certain length and which points in a certain direction. Of course, a vector doesn’t have to be a physical arrow … that’s just a helpful picture to have in your mind. Any quantity that has a magnitude (represented by the length of an arrow) and a specific direction in space (represented by the direction the arrow points) is a vector quantity.
Any quantity that has a magnitude and a specific direction in space is a vector quantity.
Seems pretty simple, right? For the most part it is. But if you think about it you’ll see that this simple idea is necessary to describe lots of things in the real world. For example, the motion of your car when you’re driving due north at 50 miles per hour can be represented by a vector pointed north with a length proportional to this speed. This vector represents the velocity of your vehicle. If you’re later on an airplane flying due north at 500 miles per hour (10 times faster), the vector representing the velocity of the airplane is 10 times longer but points in the same direction.
The bottom line is that vectors can be used to keep track of any quantity that has both a magnitude and a direction. So the change in the position of an object from one time to another as it moves through space can be represented by a vector. As can the velocity of that object over that time interval. Or its acceleration. Or even the force due to gravity that is acting on the object and thereby causing it to accelerate. All of these things are quantities that can be represented by vectors.
As you can see, the idea of a vector is inextricably linked to descriptions of real world properties of real world physical objects. And as we’ll see next time, this is closely related to one of the many real world uses of complex numbers. But more on that soon enough.
Adding, Subtracting, and Multiplying Vectors
To finish things off for today, I’d like to just touch upon a few other interesting things that you can do with vectors. First up, you can add them together. How? We’ll look at this in more detail later, but the short answer is that if you line up all the vectors from head-to-tail, the vector drawn from the tail of the first to the head of the last will be the sum of the vectors. Why is this useful? Well, imagine you’re once again traveling in that airplane heading 500 miles per hour due north. And imagine that a strong 100 mile per hour wind starts blowing from the east. How could you figure out the net velocity of the plane? You add the vectors!
Besides adding vectors, you can also subtract them and multiply them in a bunch of different ways.
There are lots of other things you can do with vectors, too. Besides adding vectors, you can also subtract them and multiply them in a bunch of different ways—you can multiply them by what are called scalar numbers (those are just ordinary everyday numbers with magnitudes but no direction associated with them), and you can even multiply one vector by another vector … in two different ways no less! (By the way, one of these ways is called a “cross product,” which is a key bit of knowledge that will help you make sense of that oh-so-funny vector joke I mentioned earlier.)
Obviously, there are a lot of details I’m leaving out. My goal today isn’t to fill you in on everything you’ll ever need to know about vectors. It’s to give you a general introduction as to what they are so that you’ll have a good foundation to work from when we do tackle these details in the future. And, of course, it’s to set up our conversation next week about one of the many awesome real world uses of complex numbers. So stay tuned for that!