Vectors???

You may have heard of vectors: they're the little arrows in physics diagrams, a package containing a size and a direction. They're important because they make it much easier to think about situations in more than one dimension, which is almost always. Mathematicians have defined what it means to "add" two vectors to get a resultant vector: you draw the two vectors so that the tail of one starts at the head of the other, and join the ends to make a new vector. You can repeat this process to add three, four or any number of vectors. (If this seems like a strange use of the word "add" you just haven't been around mathematics long enough. Alice in Wonderland was written by a mathematician. "When I use a word," Humpty Dumpty said, "it means just what I choose it to mean". You can mentally substitute some other word, if you wish, like "putogether" or something, but you'll have to get used to everyone else calling it vector addition.)

The reason why this is useful is that all sorts of things in the real world actually work this way. Velocity, force, and arcane stuff like "angular momentum" act like vectors, so we can use vector math to think about them in non-trivial ways.

The simplest physical thing that's a vector is displacement, just what the vector drawing shows. If you move from point A to point B, it's still point B regardless of how you got there. For example, if you move from one corner of a square park to the opposite corner, you can either

• Walk to a nearby corner, and then to the far corner (two straight line displacements), or
• Cut across the center of the square to the far corner (one straight line displacement)

It gets more interesting when you think about velocity. Say the park is 200 feet on a side and you walk that side in one minute. Your velocity is 200 feet/minute in the direction you are walking. Now, the same diagram as displacement can also be a velocity diagram. If you walk a little faster, about 283 feet/minute, when you are cutting across the center of the park, you will reach the far corner in one minute, because the distance to travel is about 283 feet. This means that velocity also "works" like a vector: in one minute, each velocity vector generates a displacement vector of the same direction and relative size.

The fact that velocity is a vector is really useful in thinking about real-world motion. If you throw a ball, the velocity of the ball is always changing as it moves, in a complicated way that's hard to grasp. But by taking advantage of the vector nature of velocity, we can simplify things. If the ball is moving at an angle to the horizontal, its velocity is a vector sum of a horizontal velocity and a vertical velocity. (Or, of course, lots of other possible velocities at any old angles. Those clever physicists have chosen this particular pair for a reason about to be revealed.)

The laws of physics say that the horizontal velocity stays the same (if we can neglect air resistance) because there are no forces in the horizontal direction, while the vertical velocity increases uniformly with time, because of the constant vertical force of gravity. Each component of velocity, horizontal and vertical, thus behaves simply and we can calculate the real ball velocity by just vector adding the components. This is a very standard trick used extensively in physics: divide and conquer.

If you'd like this material in more depth, try high school physics teacher Tom Henderson's Vectors, Joel Castellanos' VectorJockey shareware game or just a simple graphical Vector Adder in Java. Michael Fowler has a sophisticated/simple discussion. Then again, you may feel the the need to stamp these suckers out with a strong Vector Control Policy (a member of PestWeb!) Incidentally, the concept of vectors arrived rather late to the physics party...vectors only became standard about a hundred years ago, after most of classical mechanics had been formulated in other ways: see Prof. Crowe's book. Mathematicians and physicists think differently about many things, including vectors: get a feel by skimming this professional book review.

To get a feel for how vectors add in general, imagine you're the pilot of a spaceship traveling at a constant velocity. You fire a rocket engine for a short burst, and this changes your velocity by a vector which astronauts call "delta-v". Delta-v is in the direction exactly opposite to where your rocket blast pointed (which is completely unrelated to your current velocity, since you can point your ship's rocket in any direction). Your new velocity is the vector resultant of your original velocity and delta-v.

You need Java to do this

In the Java applet at the right, you can try this out. Click on the area to begin, then use your keyboard arrow keys to fire your rocket and create delta-v's in various directions (the size is always the same). The diagram shows the original velocity vector, the delta-v vector (in yellow) and the resultant velocity vector. Try right arrow key, then up, then left. We automagically zoom in or zoom out to fit the scale better, so the visible length of the yellow delta-v may change from keypress to keypress, but the proportions and directions are exact.

This can show one of the properties of vector addition, that the resultant can be shorter than the two vectors being added! Again, a funny kind of addition! But it makes sense if you think physically about displacement vectors. If you walk ten feet straight ahead then eight feet straight back towards your starting point, you will have a resultant displacement of only two feet. Or worse, if you walk completely around that square park we talked about before, in four straight line paths, one on each side of the square, you wind up back at your starting point. This is adding up four displacement vectors two hundred feet long and getting a resultant of zero! The math just describes the physical situation, though it sounds odd at first. This doesn't depend on it being a square, of course: can you see the vectors if you walk the park from A to B on the edges and then cut thru the center back to A?

To make this more interesting, we've made a game around it (which, by the way, uses a a lot of vector maths in the program to animate the ship!). Want to play now?

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