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Q & A: Difference between scalar and vector quantities

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Most recent answer: 09/04/2014
Q:
what is the difference between scalre and vector quantities?
- Fatima (age 16)
Pakistan
A:

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.

LeeH


(published on 05/16/2009)

Follow-Up #1: Are complex numbers scalars or vectors?

Q:
Is a "complex number (represented by a+bi, etc.)" a scalar or vector? I read before that it is a scalar number. but it seems that a complex number (intrinsically? inherently?) contains more than information of a real scalar number (and contains vector and spacial information?) because a complex number can be expressed(sits as a single point) in the complex plane(w/ real&imaginary parts). Complex numbers appear frequently in QM ("Probably the single most comprehensive use of vector spaces is in quantum mechanics.":http://van.physics.illinois.edu/qa/listing.php?id=16247) why do they appear? just for the convenience of expressing/calculating/representing/connecting vector, space, waves, etc. mathematically? or there is more intrinsic and deeper "physical(as opposed to pure math)" meaning in using complex numbers? i.e. Is a complex number an intrinsic property/part of physical world/nature/fundamental constituents(is it build-in inseparably in nature?)? or just a calculating tool or just a manifestation/facet of physical nature(a building constituent of the universe)? Coming back to the first question: So, is a complex number a scalar or vector in physics(as a study of scientific field) and in nature (what physical world is doing spontaneously)? or a complex number is (can be?) both a scalar and vector number in the study of physics/math and also? in nature? How can we observe/measure complex number/quantities in physics by direct/indirect experiments? any examples? Thank you.
- Anonymous
A:

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:

(a+bi)*(c+di)=ac-bd+(ad+bc)i.

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.

Mike W.


(published on 06/07/2013)

Follow-Up #2: what are all the numbers?

Q:
Do complex numbers include all numbers? Or there might be numbers that cannot be expressed by complex numbers (or that do not sit in complex plane)? If this is the case, why is it reasonable to assume/suspect such numbers might exist? If there cannot be such numbers beyond complex numbers(=any number can be expressed by a complex number?), why? Could you give examples (in physics and math) of numbers or quantities that cannot be a part of complex numbers if at all?
- Anonymous
A:

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.

Mike W.


(published on 09/04/2014)

Follow-up on this answer.