Constant lengths (dimensional constants) and scaling
Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is constant.
Now, scale the plane
$$x'=Cx$$
$$y'=Cy$$
Does this mean that in the new, primed coordinates, $L'=CL$? Or does $L$ stay exactly the same, $L'=L$?
I want to understand how fixed-length objects behave when the plane is scaled down (that is, when every graph of a function y=y(x) is scaled down).
It is confusing, because all variable-related objects (functions, curves, etc) in the plane must be scaled down; but $L$, being a constant, is not related to any variables, yet is defined in the plane. Does $L$ scale down also?
calculus geometry transformation constants
add a comment |
Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is constant.
Now, scale the plane
$$x'=Cx$$
$$y'=Cy$$
Does this mean that in the new, primed coordinates, $L'=CL$? Or does $L$ stay exactly the same, $L'=L$?
I want to understand how fixed-length objects behave when the plane is scaled down (that is, when every graph of a function y=y(x) is scaled down).
It is confusing, because all variable-related objects (functions, curves, etc) in the plane must be scaled down; but $L$, being a constant, is not related to any variables, yet is defined in the plane. Does $L$ scale down also?
calculus geometry transformation constants
add a comment |
Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is constant.
Now, scale the plane
$$x'=Cx$$
$$y'=Cy$$
Does this mean that in the new, primed coordinates, $L'=CL$? Or does $L$ stay exactly the same, $L'=L$?
I want to understand how fixed-length objects behave when the plane is scaled down (that is, when every graph of a function y=y(x) is scaled down).
It is confusing, because all variable-related objects (functions, curves, etc) in the plane must be scaled down; but $L$, being a constant, is not related to any variables, yet is defined in the plane. Does $L$ scale down also?
calculus geometry transformation constants
Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is constant.
Now, scale the plane
$$x'=Cx$$
$$y'=Cy$$
Does this mean that in the new, primed coordinates, $L'=CL$? Or does $L$ stay exactly the same, $L'=L$?
I want to understand how fixed-length objects behave when the plane is scaled down (that is, when every graph of a function y=y(x) is scaled down).
It is confusing, because all variable-related objects (functions, curves, etc) in the plane must be scaled down; but $L$, being a constant, is not related to any variables, yet is defined in the plane. Does $L$ scale down also?
calculus geometry transformation constants
calculus geometry transformation constants
edited yesterday
asked yesterday
user142523
17611
17611
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Hint
Take a segment $AB$ of length $L$ based on coordinates of $A$ and $B$. Compute the image of $A to A^prime$ and $B to B^prime$ by the scaling. What is the length of the segment $A^prime B^prime$?
$L=sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$; now transform $x_i ^{prime} = Cx_i$ and $y_i ^{prime} = Cy_i$, where $i in {a,b}$ and I get $L'=CL$. The confusion comes from "how you actually scale". Because in this example, $L$ seems not like a constant, but more like $L=L(x,y)$... A "constant" length should not be tied to the variables $x,y$, right? Where is my thinking wrong.
– user142523
yesterday
@user142523 $L$ is not a constant but the length is scaled by the constant of the scaling $C$.
– mathcounterexamples.net
yesterday
I was thinking of $L$ as a constant term that has a value that is constant or cannot change, because it does not contain any modifiable variables.
– user142523
yesterday
are you referring to the units of measurement, "meter". Are you saying, because the units are scaled down, this is why the dimensional constant $L$ is scaled down too?
– user142523
yesterday
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060589%2fconstant-lengths-dimensional-constants-and-scaling%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint
Take a segment $AB$ of length $L$ based on coordinates of $A$ and $B$. Compute the image of $A to A^prime$ and $B to B^prime$ by the scaling. What is the length of the segment $A^prime B^prime$?
$L=sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$; now transform $x_i ^{prime} = Cx_i$ and $y_i ^{prime} = Cy_i$, where $i in {a,b}$ and I get $L'=CL$. The confusion comes from "how you actually scale". Because in this example, $L$ seems not like a constant, but more like $L=L(x,y)$... A "constant" length should not be tied to the variables $x,y$, right? Where is my thinking wrong.
– user142523
yesterday
@user142523 $L$ is not a constant but the length is scaled by the constant of the scaling $C$.
– mathcounterexamples.net
yesterday
I was thinking of $L$ as a constant term that has a value that is constant or cannot change, because it does not contain any modifiable variables.
– user142523
yesterday
are you referring to the units of measurement, "meter". Are you saying, because the units are scaled down, this is why the dimensional constant $L$ is scaled down too?
– user142523
yesterday
add a comment |
Hint
Take a segment $AB$ of length $L$ based on coordinates of $A$ and $B$. Compute the image of $A to A^prime$ and $B to B^prime$ by the scaling. What is the length of the segment $A^prime B^prime$?
$L=sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$; now transform $x_i ^{prime} = Cx_i$ and $y_i ^{prime} = Cy_i$, where $i in {a,b}$ and I get $L'=CL$. The confusion comes from "how you actually scale". Because in this example, $L$ seems not like a constant, but more like $L=L(x,y)$... A "constant" length should not be tied to the variables $x,y$, right? Where is my thinking wrong.
– user142523
yesterday
@user142523 $L$ is not a constant but the length is scaled by the constant of the scaling $C$.
– mathcounterexamples.net
yesterday
I was thinking of $L$ as a constant term that has a value that is constant or cannot change, because it does not contain any modifiable variables.
– user142523
yesterday
are you referring to the units of measurement, "meter". Are you saying, because the units are scaled down, this is why the dimensional constant $L$ is scaled down too?
– user142523
yesterday
add a comment |
Hint
Take a segment $AB$ of length $L$ based on coordinates of $A$ and $B$. Compute the image of $A to A^prime$ and $B to B^prime$ by the scaling. What is the length of the segment $A^prime B^prime$?
Hint
Take a segment $AB$ of length $L$ based on coordinates of $A$ and $B$. Compute the image of $A to A^prime$ and $B to B^prime$ by the scaling. What is the length of the segment $A^prime B^prime$?
answered yesterday
mathcounterexamples.net
24.9k21853
24.9k21853
$L=sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$; now transform $x_i ^{prime} = Cx_i$ and $y_i ^{prime} = Cy_i$, where $i in {a,b}$ and I get $L'=CL$. The confusion comes from "how you actually scale". Because in this example, $L$ seems not like a constant, but more like $L=L(x,y)$... A "constant" length should not be tied to the variables $x,y$, right? Where is my thinking wrong.
– user142523
yesterday
@user142523 $L$ is not a constant but the length is scaled by the constant of the scaling $C$.
– mathcounterexamples.net
yesterday
I was thinking of $L$ as a constant term that has a value that is constant or cannot change, because it does not contain any modifiable variables.
– user142523
yesterday
are you referring to the units of measurement, "meter". Are you saying, because the units are scaled down, this is why the dimensional constant $L$ is scaled down too?
– user142523
yesterday
add a comment |
$L=sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$; now transform $x_i ^{prime} = Cx_i$ and $y_i ^{prime} = Cy_i$, where $i in {a,b}$ and I get $L'=CL$. The confusion comes from "how you actually scale". Because in this example, $L$ seems not like a constant, but more like $L=L(x,y)$... A "constant" length should not be tied to the variables $x,y$, right? Where is my thinking wrong.
– user142523
yesterday
@user142523 $L$ is not a constant but the length is scaled by the constant of the scaling $C$.
– mathcounterexamples.net
yesterday
I was thinking of $L$ as a constant term that has a value that is constant or cannot change, because it does not contain any modifiable variables.
– user142523
yesterday
are you referring to the units of measurement, "meter". Are you saying, because the units are scaled down, this is why the dimensional constant $L$ is scaled down too?
– user142523
yesterday
$L=sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$; now transform $x_i ^{prime} = Cx_i$ and $y_i ^{prime} = Cy_i$, where $i in {a,b}$ and I get $L'=CL$. The confusion comes from "how you actually scale". Because in this example, $L$ seems not like a constant, but more like $L=L(x,y)$... A "constant" length should not be tied to the variables $x,y$, right? Where is my thinking wrong.
– user142523
yesterday
$L=sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$; now transform $x_i ^{prime} = Cx_i$ and $y_i ^{prime} = Cy_i$, where $i in {a,b}$ and I get $L'=CL$. The confusion comes from "how you actually scale". Because in this example, $L$ seems not like a constant, but more like $L=L(x,y)$... A "constant" length should not be tied to the variables $x,y$, right? Where is my thinking wrong.
– user142523
yesterday
@user142523 $L$ is not a constant but the length is scaled by the constant of the scaling $C$.
– mathcounterexamples.net
yesterday
@user142523 $L$ is not a constant but the length is scaled by the constant of the scaling $C$.
– mathcounterexamples.net
yesterday
I was thinking of $L$ as a constant term that has a value that is constant or cannot change, because it does not contain any modifiable variables.
– user142523
yesterday
I was thinking of $L$ as a constant term that has a value that is constant or cannot change, because it does not contain any modifiable variables.
– user142523
yesterday
are you referring to the units of measurement, "meter". Are you saying, because the units are scaled down, this is why the dimensional constant $L$ is scaled down too?
– user142523
yesterday
are you referring to the units of measurement, "meter". Are you saying, because the units are scaled down, this is why the dimensional constant $L$ is scaled down too?
– user142523
yesterday
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060589%2fconstant-lengths-dimensional-constants-and-scaling%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown