On *Jeopardy! *of April 10, in the category “Dictionary of Science,” was the clue: “‘V’ is for this, defined as the speed & direction of motion of a moving body.”
I guessed “vector.” The correct response was “velocity.”
I always equated “velocity” with “speed,” which I now understand is the popular usage of the term. I’m not sure I understand what “direction” has to do with it. Does it mean that when something is changing direction while moving, it’s only the direction from the initial point to the final point that’s taken into account, although the “speed” includes the entire route, and is regardless of direction? How does that work with something that ends up back where it started? Does that mean a planet in orbit has zero velocity (ignoring the motion of the solar system as a whole)? So “velocity” and “vector” are synonymous; or is there a distinction?
Velocity is speed AND direction combined. So if you are going around a curve your speed may stay the same but your velocity is constantly changing. A speed would be something like 100 km/h. A velocity would be something like 100 km/h heading north.
Technically velocity is a vector which means it has a magnitude and a direction; 55 mph northeast. (It does not actually have a location though). Speed is the magnitude of the vector, 55 mph.
So velocity is a vector but not a synonym for vector. There are other vectors such as force. A ten pound weight is pushing with a force of ten pounds straight down.
One could define going 100 m/s due north as +100 m/s. Then going 100 m/s due south would be -100 m/s. Of course, one can’t define going due east or west in this system. For that you would need a two-dimensional vector (e.g. east is [100, 0] and south is [0, -100]).
Yes, that’s because speed is the MAGNITUDE of the velocity vector. The layman uses the two terms interchangeably, but they are technically not the same. It only makes a difference to science and engineering folks.
A vector has magnitude, direction, and sign. If the magnitude happens to be a speed, then the vector is specifically a velocity vector (as was the case on Jeopardy), often noted with the symbol V - but there are many other kinds of vectors, such as force, torque, displacement, etc.
In a two or more dimensional space, the only possible meaning absolute value could have is magnitude, so equating the concepts is perfectly correct. Yes, it relies on context to disambiguate, but communication always depends on context, and the basic ideas are so closely linked anyway that trying to draw a distinction is actually more confusing.
So, to tie it all together: Velocity is a directed magnitude, which means it can be represented mathematically by a vector. Speed is the common term for the magnitude of a velocity, which means it can be represented mathematically as a scalar, which is what you call numbers like 0.5 and 88 once you bring vectors into it.
Now, let me wrap this up before the divergence of velocities around me becomes too great.
I assumed the point would be that it would break down if you had 2 dimensional velocity rather than one dimensional with a modifier. If the velocity is 3+4i, the term absolute value (3+4i) isn’t going to tell you the magnitude. You’ll need the Pythagorean theorem to find the magnitude is 5.
But, in that context, absolute value means magnitude. When we write |a+bi| we mean \sqrt(a[sup]2[/sup]+b[sup]2[/sup]). As Derleth points out, in two dimensions and higher, the most useful meaning of “absolute value” is magnitude.
A vector quantity is, most basically, something that takes more than one number to describe. Often, especially in the context of physical science, it’s pictured as a directed line segment: think of an arrow with a specific length and direction. By contrast, a scalar quantity is one that only takes a single number to describe. So, speed is a scalar quantity: it’s the single number that tells how fast something is going (in miles per hour or meters per second or whatever). Velocity is a vector quantity: it tells not only how fast you’re going, but it what direction. This is an important distinction in physics. An object remains at constant velocity unless acted on by an outside force (which might include gravity or friction); it takes an outside force to change an object’s speedordirection of motion.
(If you’re driving around a circular racetrack at a constant speed, your velocity is constantly changing even though your speed isn’t.)
Maybe part of the OP’s problem is that, in English, speed and velocity are unique. No other vector has an explicitly different term for its scalar equivalent.
I would say that the difference is that a vector is a mathematical construct. It’s any value that combines both a value and a direction. Velocity is described with a vector, just as speed is described with a number (or, more accurately, two numbers - distance and time).
Yeah, but the sign is part of the direction compontent. It basically means “rotate the following vector by 180 degrees”. So it is sufficient to say a vector is just a value with both a magnitude and direction.
Right. you can change a vector’s magnitude without changing the direction, or change the direction without changing the magnitude, but, to the extent you can define a ‘sign’ for a vector, you can’t change it without changing either the magnitude or the direction.
So, it’s more accurate to say “A vector has magnitude and direction”
