Rotation of $e_1 in mathbb{R}^n$ in angles along the axis
$begingroup$
I have the vector $e_1=(1,0,...,0)^T$ in $mathbb{R}^n$.
I would like to rotate it by angle $theta_2$ along axis $x_2$, resulting in the vector $r_1 = (cos(theta_2),sin(theta_2),0,...,0)^T$.
Continuing these rotation by angle $theta_i$ along axis $x_i$ for $2 < i leq n$, what would be the form of the resulting vector?
I couldn't find a general rotation formula.
linear-algebra rotations angle
asked Jan 11 at 15:30
catch22catch22
1,3361122
$endgroup$
add a comment |
$begingroup$
I have the vector $e_1=(1,0,...,0)^T$ in $mathbb{R}^n$.
I would like to rotate it by angle $theta_2$ along axis $x_2$, resulting in the vector $r_1 = (cos(theta_2),sin(theta_2),0,...,0)^T$.
Continuing these rotation by angle $theta_i$ along axis $x_i$ for $2 < i leq n$, what would be the form of the resulting vector?
I couldn't find a general rotation formula.
linear-algebra rotations angle
asked Jan 11 at 15:30
catch22catch22
1,3361122
$endgroup$
add a comment |
$begingroup$
I have the vector $e_1=(1,0,...,0)^T$ in $mathbb{R}^n$.
I would like to rotate it by angle $theta_2$ along axis $x_2$, resulting in the vector $r_1 = (cos(theta_2),sin(theta_2),0,...,0)^T$.
Continuing these rotation by angle $theta_i$ along axis $x_i$ for $2 < i leq n$, what would be the form of the resulting vector?
I couldn't find a general rotation formula.
linear-algebra rotations angle
asked Jan 11 at 15:30
catch22catch22
1,3361122
$endgroup$
I have the vector $e_1=(1,0,...,0)^T$ in $mathbb{R}^n$.
I would like to rotate it by angle $theta_2$ along axis $x_2$, resulting in the vector $r_1 = (cos(theta_2),sin(theta_2),0,...,0)^T$.
Continuing these rotation by angle $theta_i$ along axis $x_i$ for $2 < i leq n$, what would be the form of the resulting vector?
I couldn't find a general rotation formula.
linear-algebra rotations angle
linear-algebra rotations angle
asked Jan 11 at 15:30
catch22catch22
1,3361122
asked Jan 11 at 15:30
catch22catch22
1,3361122
asked Jan 11 at 15:30
catch22catch22
1,3361122
asked Jan 11 at 15:30
catch22catch22
1,3361122
asked Jan 11 at 15:30
catch22catch22
1,3361122
1,3361122
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The reason that there is no general formula, is that the rotation that you are seeking is not unique. To rotate about the $x_2$ axis you need to identify a plane which is normal to $x_2$. For $n > 3$ this plane in not unique. There are many planes perpendicular to $x_2$ which live in the space spanned by ${x_1,, x_3,, ...,, x_n}$.
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
$endgroup$
add a comment |
$begingroup$
What I am assuming you want is to generate the rotation matrices in n dimensions, note that you do not rotate around an axis anymore but rather in a specific plane (more than 1 axis is orthogonal to it). Just as an example, in 4d you can rotate around zw, yw, xw, yz, xz, xy. The rotation matrices for these have 1s on the diagonal around the axes you are rotating and the standard 2d rotation (possibly flipped) in the other 2 spots. For examples for a rotation around zw you have $a_{1,1} = cos theta_1 = a_{2,2}, - a_{1,2} = sin theta = a_{2,1}, a_{3,3}=a_{4,4} = 1$, all other elements of the matrix are 0. To get your vector you can just multiply it with the matrices one after another.
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
$endgroup$
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%2f3069961%2frotation-of-e-1-in-mathbbrn-in-angles-along-the-axis%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The reason that there is no general formula, is that the rotation that you are seeking is not unique. To rotate about the $x_2$ axis you need to identify a plane which is normal to $x_2$. For $n > 3$ this plane in not unique. There are many planes perpendicular to $x_2$ which live in the space spanned by ${x_1,, x_3,, ...,, x_n}$.
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
$endgroup$
add a comment |
$begingroup$
The reason that there is no general formula, is that the rotation that you are seeking is not unique. To rotate about the $x_2$ axis you need to identify a plane which is normal to $x_2$. For $n > 3$ this plane in not unique. There are many planes perpendicular to $x_2$ which live in the space spanned by ${x_1,, x_3,, ...,, x_n}$.
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
$endgroup$
add a comment |
$begingroup$
The reason that there is no general formula, is that the rotation that you are seeking is not unique. To rotate about the $x_2$ axis you need to identify a plane which is normal to $x_2$. For $n > 3$ this plane in not unique. There are many planes perpendicular to $x_2$ which live in the space spanned by ${x_1,, x_3,, ...,, x_n}$.
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
$endgroup$
The reason that there is no general formula, is that the rotation that you are seeking is not unique. To rotate about the $x_2$ axis you need to identify a plane which is normal to $x_2$. For $n > 3$ this plane in not unique. There are many planes perpendicular to $x_2$ which live in the space spanned by ${x_1,, x_3,, ...,, x_n}$.
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
answered Jan 17 at 0:12
TpofofnTpofofn
3,5971427
3,5971427
add a comment |
add a comment |
$begingroup$
What I am assuming you want is to generate the rotation matrices in n dimensions, note that you do not rotate around an axis anymore but rather in a specific plane (more than 1 axis is orthogonal to it). Just as an example, in 4d you can rotate around zw, yw, xw, yz, xz, xy. The rotation matrices for these have 1s on the diagonal around the axes you are rotating and the standard 2d rotation (possibly flipped) in the other 2 spots. For examples for a rotation around zw you have $a_{1,1} = cos theta_1 = a_{2,2}, - a_{1,2} = sin theta = a_{2,1}, a_{3,3}=a_{4,4} = 1$, all other elements of the matrix are 0. To get your vector you can just multiply it with the matrices one after another.
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
$endgroup$
add a comment |
$begingroup$
What I am assuming you want is to generate the rotation matrices in n dimensions, note that you do not rotate around an axis anymore but rather in a specific plane (more than 1 axis is orthogonal to it). Just as an example, in 4d you can rotate around zw, yw, xw, yz, xz, xy. The rotation matrices for these have 1s on the diagonal around the axes you are rotating and the standard 2d rotation (possibly flipped) in the other 2 spots. For examples for a rotation around zw you have $a_{1,1} = cos theta_1 = a_{2,2}, - a_{1,2} = sin theta = a_{2,1}, a_{3,3}=a_{4,4} = 1$, all other elements of the matrix are 0. To get your vector you can just multiply it with the matrices one after another.
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
$endgroup$
add a comment |
$begingroup$
What I am assuming you want is to generate the rotation matrices in n dimensions, note that you do not rotate around an axis anymore but rather in a specific plane (more than 1 axis is orthogonal to it). Just as an example, in 4d you can rotate around zw, yw, xw, yz, xz, xy. The rotation matrices for these have 1s on the diagonal around the axes you are rotating and the standard 2d rotation (possibly flipped) in the other 2 spots. For examples for a rotation around zw you have $a_{1,1} = cos theta_1 = a_{2,2}, - a_{1,2} = sin theta = a_{2,1}, a_{3,3}=a_{4,4} = 1$, all other elements of the matrix are 0. To get your vector you can just multiply it with the matrices one after another.
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
$endgroup$
What I am assuming you want is to generate the rotation matrices in n dimensions, note that you do not rotate around an axis anymore but rather in a specific plane (more than 1 axis is orthogonal to it). Just as an example, in 4d you can rotate around zw, yw, xw, yz, xz, xy. The rotation matrices for these have 1s on the diagonal around the axes you are rotating and the standard 2d rotation (possibly flipped) in the other 2 spots. For examples for a rotation around zw you have $a_{1,1} = cos theta_1 = a_{2,2}, - a_{1,2} = sin theta = a_{2,1}, a_{3,3}=a_{4,4} = 1$, all other elements of the matrix are 0. To get your vector you can just multiply it with the matrices one after another.
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
answered Jan 17 at 1:08
lightxbulblightxbulb
69219
69219
add a comment |
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.
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%2f3069961%2frotation-of-e-1-in-mathbbrn-in-angles-along-the-axis%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
