Can you travel negative distance?

Can you go on a -60 mile journey? What about trying to drive at -40mph? Wouldn’t doing either of these things be breaking some law of the universe and send you back in time or something? Fortunately for you, probably not, because measuring quantities like these using negative numbers is pretty common when we’re using vectors.

To see why vectors are useful, let’s imagine you’re a delivery driver who’s out delivering parcels. You’re new to the job, so you’ve only been given a single street to deal with, but unfortunately none of the buildings have numbers on them! (maybe that’s why the last guy quit…). How are you supposed to know where to deliver the packages?

Luckily, you’ve been sent out with the GPS Address Decoder 3000: Simply punch in an address and it’ll use fancy GPS satellites to tell you exactly how far you need to move to get there! It’s state of the art technology! Or at least it probably was in 1992…

…Either way, it’s all you’ve got, so it’ll have to do. To make our job easier, let’s simplify this street as a grid. Each building is roughly the same distance apart, so let’s call that distance 1 unit:

It’s time for your first delivery! Let’s see, the first package is addressed to the house… “Ahead in Red”? Let’s give your gadget a try:

…Well, you can’t say it isn’t concise. Let’s move 6 along and make your delivery!

Great, that wasn’t so hard. Where’s the next package going? “The Blue House”? Hmm, no marks for creativity with that one. Let’s ask the decoder:

That seems easy enough… But wait, isn’t there a problem?

Move 3 in which direction?

This is no good - how will you know where this package is going now? The Blue House isn’t helpful – they’re both blue houses! And what’s this weird vector mode on our decoder? Let’s turn it on and see if it can help:

Huh? Now it’s telling us to move -3? How are we supposed to move a negative amount? Perhaps it’ll make more sense if we number the buildings on your street:

Now it’s a bit clearer, right? This is just a number line! If we’re at position 6, then moving -3 just means moving to position 3, because 6 – 3 = 3 !

This is how vectors represent direction – a positive number means move one way, and a negative number means move the other. If our decoder had still told us to move positive 3 once we activated vector mode, then we would have moved the other way and ended up at position 6+3: position 9:

This also demonstrates the difference between distance and displacement. In both of the above scenarios, we travelled a distance of 3, and yet we ended up in different spots! This is because distance only describes the size, or magnitude, of the amount we travelled – it doesn’t care about direction! This is why distance isn’t a vector, and also why we can’t travel a negative distance. What would it mean to move an amount that was less than zero? It doesn’t make sense!

If we need to take direction into account, we have to use displacement. This is what the decoder is displaying when its vector mode is activated! Displacement is a vector, because it has the two qualities of a vector:

• A magnitude (this is your distance!)
• A direction

This is why displacement can be negative: the negative sign doesn’t mean it has a magnitude of less than zero, it just means the magnitude is pointing in the opposite direction! When we talk about real-world quantities, we often encounter vector-magnitude pairs like displacement and distance. Another common one is velocity and speed: speed just tells you how fast something is moving, but velocity can tell you which direction its moving in as well!

It’s now your next day at work, and since you did so well yesterday, you’ve been given two streets to deliver to:

What’s your first instruction?

Now your displacement is -2. That just means you moved a distance of 2 in the negative direction. It’s useful to note that which direction is positive and which direction is negative is completely arbitrary, we could easily do it the other way around if we felt like it! The only important thing is that both you and the decoder agree which is which, so you don’t start moving the wrong way.

Ok, you’re getting the hang of this! Let’s see where this next package is go-

Huh? It has a note on it:

Great. Your second day and you’re already being given extra work – now you’ve got the whole block to deliver to!

Where have you got to go now? Let’s head back to 0 – that should make the instructions easier to understand.

Well, that doesn’t seem too hard… Wait! The machine’s switched itself off of vector mode! (this is why you shouldn’t be relying on 30-year-old tech…). Let’s switch it back on:

…What the heck? What’s $\left(\begin{array}{c}3\\ 4\end{array}\right)$ supposed to mean? Why are there two numbers now? The decoder told you a second ago that you needed to move a distance of 5, so why are NEITHER of these new numbers a 5? Let’s look at a grid of the block:

That sure is a lot more houses. We know that your next house is a distance of 5 units away, so let’s consider that first. How many possible locations could the next house be at?

When you were delivering before, you were moving in a single dimension, so there were 2 possible directions in which you could travel - forwards and backwards. This meant there were 2 locations that were a distance of 5 away. Now that you have to deliver to an entire block of houses, you’ll have to move side-to-side as well, meaning that you’ll have to move in 2 dimensions (the grid above labels this second dimension by adding a second, vertical number line). So, if 1 dimension means 2 possible locations, then perhaps 2 dimensions gives us 4?

That seems right, right? Actually, wrong! The 4 places marked above certainly are possible locations, but there are a bunch more we’ve missed out! Look:

The circle marks every point on the grid that’s a distance of 5 away from you – that’s a lot more than 4! (fun fact: a set of points all of equal distance from a certain place is actually the definition of a circle!). Even if we assume that there are only houses at the points where the grid squares cross, that’s still 12 possible locations! This point at the coordinate (3,4), for example, was missed out entirely:

…Hang on, (3,4)? Doesn’t that look familiar?

It’s almost the same as the vector your decoder gave you! In fact, if you start from 0, moving by the vector $\left(\begin{array}{c}3\\ 4\end{array}\right)$ will take you to the coordinate (3,4)!

This is because, in the same way a coordinate gives a horizontal position and a vertical position \((x, y)\), a vector gives a horizontal magnitude and direction, and a vertical magnitude and direction, $\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$. We need to move in 2 dimensions, so of course our vector needs 2 numbers – one to describe each of them! We can recover the overall magnitude of the vector (the total distance of 5) using Pythagoras’ Theorem:

$$\sqrt{3^2+4^2}=5$$

Wait, Pythagoras? Isn’t that to do with triangles? Yep! We can use it here because the two numbers that form the vector actually describe the two sides of a right-angled triangle:

Let’s try another delivery:

Ok, so this means you need to travel 4 units in the positive horizontal direction, and 2 units in the positive vertical direction, right?

Sure does! But wait, you moved $\left(\begin{array}{c}4\\ 2\end{array}\right)$, but you’re not at the coordinate (4,2): you’re at (7,6)! Why’s that?

It’s because you didn’t start at (0,0) this time – you were already at (3,4), remember? Your new vector got added on to your previous one:

$$\left(\begin{array}{c}3\\ 4\end{array}\right)+\left(\begin{array}{c}4\\ 2\end{array}\right)=\left(\begin{array}{c}7\\ 6\end{array}\right)$$

Huh. Seems as though we can add vectors together just like regular numbers: 3+4=7, 4+2=6. That’s useful!

Just one more delivery left for today! Where’s it going?

Now we know we can just add vectors together to get our new coordinate, let’s see if we can figure out where you’ll end up without looking at the grid at all! You’re currently at position (7,6), so:

$$\left(\begin{array}{c}7\\ 6\end{array}\right)+\left(\begin{array}{c}0\\ -9\end{array}\right)=\left(\begin{array}{c}7\\ -3\end{array}\right)$$

Do you end up at coordinate (7,-3)?

Yep!

So to summarise:

• Vectors are objects that contain both a magnitude and a direction
• How many numbers, or components, we need for our vector depends on how many dimensions we’re working in – 2 dimensions requires vectors with 2 components!
• Vectors can have negative components – the negative is contributing to the direction part, not the magnitude part!
• We can add vectors together similarly to how we add non-vectors

and finally…

• Delivery companies need to stop using technology they bought in the early nineties