Vector Calculus in 1D: ± to Show Magnitude?

In summary: In 1D, the gravitational acceleration vector and the acceleration vector, both pointing down, are the same. The equation ##T-mg=ma## does not say that the acceleration of the mass is -g. It says that the acceleration of the mass minus that of gravity is ma.In summary, the conversation discusses the notation for representing vector quantities in one dimension and the potential ambiguity that can arise in this representation. It is important to maintain a consistent choice of positive direction in order to avoid confusion. Furthermore, the use of the same symbol for both speed and velocity can also lead to misunderstandings. This is why textbooks often distinguish between the two and define speed as the magnitude of velocity. Another potential issue arises in the case
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MatinSAR
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[mentor's note - moved from one of the homework help forums]

Homework Statement:: It's a question.
Relevant Equations:: Vector calculus.

Is it true to say that in one dimension I can show vector quantities using ±number instead of a vector?
± can show possible directions in one dimension and that number shows magnitude of quantity.

Thanks.
 
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Yes!
 
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vanhees71 said:
Yes!
Thank you for your time.

@malawi_glenn Thank you for your help.
 
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MatinSAR said:
Is it true to say that in one dimension I can show vector quantities using ±number instead of a vector?
± can show possible directions in one dimension and that number shows magnitude of quantity.
Consider where this is coming from. A 3D vector is written as $$\mathbf{A}=A_x~\mathbf{\hat x}+A_y~\mathbf{\hat y}+A_z~\mathbf{\hat z}$$ where the components ##A_x##, ##A_y## and ##A_z## can be positive or negative. Of course, to write down a vector in this manner, you must have a coordinate system with unit vectors already defined.

In the special case ##A_y=A_z=0##, you have a 1D vector which is formally written as $$\mathbf{A}=A_x~\mathbf{\hat x}.$$ Note that ##A_x## can still be positive or negative.

Informally, the vector notation and the single unit vector are dropped, but the choice of which way the single axis is positive must remain. Ambiguity may arise when one sees in 1D something like ##v = -2~##m/s. Is this a vector equation or not? There is no ambiguity in the vector equation ##\mathbf{v}=(-2)~\mathbf{\hat x}~##m/s or in scalar equation ##v_x=-2~##m/s.

Dropping the vector notation and the subscript (a widespread practice in 1D) conflates the two and could be a source of confusion to people who don't know what's "under the hood." It looks like you were confused about this yourself and that is why you asked your question.
 
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There is also an inbuilt ambiguity due to speed and velocity having the same symbol ##v## in that case
 
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malawi_glenn said:
There is also an inbuilt ambiguity due to speed and velocity having the same symbol ##v## in that case
Thank you for your help.
kuruman said:
Consider where this is coming from. A 3D vector is written as $$\mathbf{A}=A_x~\mathbf{\hat x}+A_y~\mathbf{\hat y}+A_z~\mathbf{\hat z}$$ where the components ##A_x##, ##A_y## and ##A_z## can be positive or negative. Of course, to write down a vector in this manner, you must have a coordinate system with unit vectors already defined.

In the special case ##A_y=A_z=0##, you have a 1D vector which is formally written as $$\mathbf{A}=A_x~\mathbf{\hat x}.$$ Note that ##A_x## can still be positive or negative.

Informally, the vector notation and the single unit vector are dropped, but the choice of which way the single axis is positive must remain. Ambiguity may arise when one sees in 1D something like ##v = -2~##m/s. Is this a vector equation or not? There is no ambiguity in the vector equation ##\mathbf{v}=(-2)~\mathbf{\hat x}~##m/s or in scalar equation ##v_x=-2~##m/s.

Dropping the vector notation and the subscript (a widespread practice in 1D) conflates the two and could be a source of confusion to people who don't know what's "under the hood." It looks like you were confused about this yourself and that is why you asked your question.
Thank you for your detailed answer. I was confused when I start to reread about motion along a straight line in "Fundamentals of Physics (Textbook by David Halliday)".
 
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  • #7
malawi_glenn said:
There is also an inbuilt ambiguity due to speed and velocity having the same symbol ##v## in that case
Textbooks usually distinguish typographically between speed (v) and velocity (v or ##\vec v##), and define speed as the magnitude of the velocity. I.e., ##v = |\vec v|## = |v|.
 
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A related problem with writing 1D vector equations appears in the accelerating hanging mass. Students correctly write down the equation ##~T-mg=ma.## When they are asked to find the tension in the case of a hanging mass accelerating down with acceleration 2 m/s2, they correctly substitute -2 m/s2 for ##a## in the equation but are puzzled to find out that the substitution of -9.8 m/s2 for ##g## gives the wrong answer.
 
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1. What is vector calculus in 1D?

Vector calculus in 1D is a branch of mathematics that deals with the study of vectors and their properties in one-dimensional space. It involves the use of mathematical operations such as differentiation and integration to analyze and solve problems involving one-dimensional vectors.

2. What is the purpose of using vectors in 1D?

Vectors in 1D are used to represent physical quantities that have both magnitude and direction. They are useful in many fields of science, such as physics and engineering, for describing motion, forces, and other physical phenomena.

3. How do you calculate the magnitude of a vector in 1D?

The magnitude of a vector in 1D can be calculated using the Pythagorean theorem, which states that the magnitude is equal to the square root of the sum of the squares of the vector's components. In 1D, this reduces to simply taking the absolute value of the vector's single component.

4. What does the ± symbol mean in vector calculus in 1D?

The ± symbol in vector calculus in 1D represents the two possible directions of a vector in one-dimensional space. For example, if a vector has a magnitude of 5 and is represented as ±5, it can either be pointing in the positive or negative direction along the one-dimensional axis.

5. How is vector calculus in 1D applied in real-world situations?

Vector calculus in 1D is applied in many real-world situations, such as calculating the velocity and acceleration of an object moving in one-dimensional space, determining the forces acting on an object, and analyzing the flow of fluids or electricity in a one-dimensional system. It is also used in mathematical modeling and computer simulations in various fields of science and engineering.

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