- #1
Amik
- 38
- 3
- Homework Statement
- In the solution
- Relevant Equations
- W=FD
I am so confused.If F and d are both vector quantity.How come W is a scalar quantity?
No, this is incorrect.Amik said:I think scalar quantity can only be positive?
It is a very bad quality picture. Please take the habit of writing out relevant quotes.Amik said:The picture I sent you in the first picture said this
Sorry, but this is not writing out the relevant quote. Writing out the quote means citing it word by word. What you wrote is your interpretation.Amik said:I am sorry.The last line of the first picture said work is a scalar quantity.
You thinking so does not make it so. You are simply wrong about this. Correcting this misconception is fundamental for your future understanding.Amik said:I think scalar quzntity can only be positive.
So what? Particular scalar quantities may fall into some restricted range because of their definitions. This does not mean that all scalars do. For example, work is a scalar quantity that may be negative. Electric potential is a scalar quantity that may be negative. There are many other examples.Amik said:Because for example, speed is a scalar quantity and it has no direction.And speed can only be positive.
If measured in °C or °F, yes. If measured in K (as is more typical in physics), it is positive.Amik said:It is like temperature it is scalar quantity but it can either positive or negative.
A scalar can be both positive and negative. The numbers are scalars, and "2" is positive, "-5" is negative.Amik said:I think scalar quantity can only be positive?and work is scalar quantity?
You are overlooking this part of the quoted textehild said:A scalar can be both positive and negative. The numbers are scalars, and "2" is positive, "-5" is negative.
The magnitude or absolute value is positive or zero.
This is what has misled Amik. It should say that scalars have a sign and a magnitude but not a direction.Amik said:"... It has a magnitude but not a direction"
As I noted in post #17, the root of the error seems to be in the book's statement that scalars have only magnitude. I have since found this to be almost ubiquitous on the web (even NASA) and taken it upon myself to contact the site managers where possible. So far I have had two responses; both thanked me and promised correction.archaic said:This is a matter of notation. Some textbooks would use a bold character to denote a vector and a normal character to denote scalars. Hence, the force vector would be ##\mathbf F## and its magnitude (which is a scalar quantity) would be ##F## or ##\mathrm F##. Other textbooks, however, might denote vectors like this ##\vec F## and their magnitude as ##||\vec F||##, ##F## or ##\mathrm F##.
From the text before the equation you have, it is clear that your textbook uses the second notation.
Not at all. The -3 is the scalar coefficient of the unit vector.kuruman said:You would probably answer, "No, the negative sign indicates the direction, which is to the left. The scalar quantity is just 3 m/s without the direction
kuruman said:Suppose now someone said to you the velocity of an object is ##v=-3~\mathrm{m/s}##. Question: is ##-3~\mathrm{m/s}## a scalar quantity?
Yes it is. I am attempting to explain the origin of the misconception. When one writes "The velocity of the object is v = -3 m/s", the "-3 m/s" is the coefficient of a unit vector which is not explicitly shown. An expert understands how to interpret that. A beginner might think that the negative sign is actually the unit vector because after all it indicates direction "to the left". A scalar has no direction therefore to get the scalar that multiplies direction in this vector expression, one has to strip the negative sign.haruspex said:Not at all. The -3 is the scalar coefficient of the unit vector.
I don't see it that way. There is nothing scalar-like in ##\vec v=(-3~\mathrm{m/s}) ~\hat x##. It's the removal of the symbols denoting vector quantities (for convenience?) that makes the expression scalar-like.PeroK said:If you constrain motion to one dimension, you restrict vector quantities like velocity and make them scalar-like.
It is scalar-like because that vector only requires one number to specify it.kuruman said:I don't see it that way. There is nothing scalar-like in ##\vec v=(-3~\mathrm{m/s}) ~\hat x##. It's the removal of the symbols denoting vector quantities (for convenience?) that makes the expression scalar-like.
Hmm.. well, you can construct a 1D vector space from the reals (or complex numbers, or any field) in a trivial way, but it does involve a construction. It does not mean the reals are a vector space.PeroK said:Real (and Complex) numbers are both vectors and scalars
I would say the problem is the common description that scalars only have magnitude, hence are nonnegative. Whether having a sign is equivalent to having direction is debatable. E.g. charge has sign, but does it have direction?vela said:The problem seems to be the common description that scalars don't have direction
##\mathbb{R}^n## is a vector space for all ##n##, including ##n = 1##. No "construction" is needed. They satisfy all the axioms of a vector space, with ##\mathbb{R}## as the field of scalars.haruspex said:Hmm.. well, you can construct a 1D vector space from the reals (or complex numbers, or any field) in a trivial way, but it does involve a construction. It does not mean the reals are a vector space.
Strictly speaking, ##\mathbb{R}^n## is just the set of n-tuples {(x1,..,xn)}. To make it a vector space you have to specify two sets, the set of vectors and the field of scalars, then define how the vector elements add and what the product of a scalar and a vector is.PeroK said:##\mathbb{R}^n## is a vector space for all ##n##, including ##n = 1##. No "construction" is needed. They satisfy all the axioms of a vector space, with ##\mathbb{R}## as the field of scalars.
The original discussion was regarding ##\mathbb R## itself, not ##\mathbb R^n##. Any field is a vector space over itself without any additional structure as it already has addition and multiplication by scalars. I therefore do not think it makes sense to claim that ##\mathbb R## is not a vector space - it has all of the required properties. That does not mean it cannot also be made into a vector space over a different field, but in itself it has all of the required properties to be called a vector space.haruspex said:Strictly speaking, ##\mathbb{R}^n## is just the set of n-tuples {(x1,..,xn)}. To make it a vector space you have to specify two sets, the set of vectors and the field of scalars, then define how the vector elements add and what the product of a scalar and a vector is.
That we commonly think of ##\mathbb{R}^n## as the natural vector space ##\{\mathbb{R}^n, \mathbb{R}\}## is just a convenient shorthand.
Is there only one way to do that with ##\mathbb{R}^n## as a set of vectors over the field ##\mathbb{R}##?
What if I take the n-tuples ##\mathbb{C}^n## over the field ##\mathbb{R}##? Doesn't that make a vector space in the obvious way?
haruspex said:Strictly speaking, ##\mathbb{R}^n## is just the set of n-tuples {(x1,..,xn)}.
##\mathbb{R}## is defined (as I understand it) not just as a set but all the operations and axioms necessary to be a field. I accept that one could also define something else with the same name (a pun) which is a vector space constructed with the elements of the set underlying ##\mathbb{R}## as its vectors, over the field ##\mathbb{R}##, with all the addition and multiplication rules defined in the obvious way.PeroK said:What is ##\mathbb{R}## for that matter? Strictly speaking it has no properties until you define it by some sort of construction. It's a set of equivalence classes of Cauchy sequences of rationals if you want to start at the beginning.
There's no reason arbitrarily to decide that ##\mathbb{R}## needs "construction" specifically to be a vector space, but no construction to be a field of scalars.
So the idea is that one is using R as the set of elements of the vector space and R as the set of scalars.haruspex said:##\mathbb{R}## is defined (as I understand it) not just as a set but all the operations and axioms necessary to be a field. I accept that one could also define something else with the same name (a pun) which is a vector space constructed with the elements of the set underlying ##\mathbb{R}## as its vectors, over the field ##\mathbb{R}##, with all the addition and multiplication rules defined in the obvious way.
But it is an interesting question as to whether there is a nonobvious construction that makes a different vector space.
Those are choices that nail it down.jbriggs444 said:With ordinary real addition for the addition of two vectors and ordinary real multiplication for the product of a vector and a scalar.
Yes, work is a vector quantity in physics. This means that it has both magnitude and direction.
In physics, work is defined as the product of the force applied on an object and the displacement of the object in the direction of the force.
Yes, work can be negative in physics. This occurs when the force and displacement are in opposite directions, resulting in a negative value for work.
The unit of measurement for work in physics is joules (J). It can also be expressed in other units such as newton-meters (N*m) or kilogram-meters squared per second squared (kg*m^2/s^2).
In physics, work and energy are closely related concepts. Work is the transfer of energy from one system to another, and the amount of work done on an object is equal to the change in its energy.