Difference Between Scalar and Vector Quantities
Most recent answer: 05/16/2009
- Fatima (age 16)
A scalar quantity is a one dimensional measurement of a quantity, like temperature, or mass. A vector has more than one number associated with it. A simple example is velocity. It has a magnitude, called speed, as well as a direction, like North or Southwest or 10 degrees west of North. You can have more that two numbers associated with a vector. For example you can add a height dimension to velocity and say, for example, ' I am going uphill at a 5 degree slope in the Northeast direction'. Vectors are frequently broken down into their components along an orthogonal coordinate system, like the x and y axes. So you can say the y-component of my speed is 3 km/sec and the x-component of my speed is 4 km/sec. The magnitude, or speed is the square root of the sum of the individual components, 5 in this case. The direction with respect to the x-axis would be given by the arctangent of Vy / Vx or 36.9 degrees.
(published on 05/16/2009)
Follow-Up #1: Are complex numbers scalars or vectors?
This is a nice follow-up.
The set of complex numbers is indeed a 2D vector space if you pay attention only to what happens when you add the numbers. However, what makes complex numbers special is their multiplication rule:
This is very different from the generic 2D vector dot product, ac+bd, which gives a number, not another vector. This complex multiplication rule makes the complex numbers form a mathematical field (), something you can then multiply vectors by to get the same type of vector you started with. In other words, it lets complex numbers be used as scalars, just as you use real numbers. In physics we routinely use them as scalars, in that they don't change under spatial rotations, unlike say standard 3-vectors or 4-vectors.
You can measure things like ac magnetic susceptibility that we represent, with good reason, as complex numbers. It's not necessary to use that representation, but it's very convenient. I think what you're probably more interested in is the complex quantum wavefunction. With some awkwardness, it could be represented without explicit complex numbers, but whatever mathematical expression was used would end up being exactly equivalent. The wavefunction isn't quite measurable, since there's always an arbitrary absolute complex phase factor. (Relative phases are defined.)
The role of vector spaces in quantum mechanics goes far deeper than the mere use of complex numbers. The quantum states themselves form a (generally infinite-dimensional) vector space. Physical measurables, like energy, are represented by linear operators on this vector space, like matrices on finite-dimensional vector spaces. The time-dependence operator is length-preserving, so it's just a rotation in the abstract vector space. Once you start thinking of quantum mechanics in terms of these state vectors, you'll get hooked.
As for the philosophical questions about intrinsic reality etc., we have little to say.
(published on 06/07/2013)
Follow-Up #2: what are all the numbers?
If you start with simple counting numbers, there are good reasons to add more types. For example, you want negatives to allow you to solve equations like x+5=4. Those negatives have an intuitive meaning. The same goes for fractions, needed to solve equations like 5x=4. Continuing with this process of inventing new numbers to solve equations gets you to various irrational numbers like Sqrt(2) and then to complex numbers like Sqrt(-1). Including all the limits of sequences of these numbers gives you the full set of complex numbers. Then it turns out you're done- you have all the solutions to all your equations. It's traditional to use the word "numbers" to name all these extensions of our counting numbers.
There are plenty of other mathematical objects, however. These include, for examples, matrices and quaternions. These other objects aren't called numbers, however.
(published on 09/04/2014)