Mdthbs

Narset, Parter of Veils interaction with Aria of Flame

Why did House of Representatives need to condemn Trumps Tweets?

What is the German equivalent of the proverb 水清ければ魚棲まず (if the water is clear, fish won't live there)?

What is the reason for cards stating "Until end of turn, you don't lose this mana as steps and phases end"?

Why does Canada require bilingualism in a lot of federal government posts?

How can I kill my goat?

Was the Psych theme song written for the show?

Exploiting the delay when a festival ticket is scanned

Argand formula and more for quaternions?

Do the books ever say oliphaunts aren’t elephants?

Newton's cradles

What would the United Kingdom's "optimal" Brexit deal look like?

Complaints from (junior) developers against solution architects: how can we show the benefits of our work and improve relationships?

How does a poisoned arrow combine with the spell Conjure Barrage?

Story about separate individuals coming together to make some supernatural universe power

Why are we moving in circles with a tandem kayak?

Why is my fluorescent tube orange on one side, white on the other and dark in the middle?

Just how much information should you share with a former client?

Why was the LRV's speed gauge displaying metric units?

Why put copper in between battery contacts and clamps?

Why did Windows 95 crash the whole system but newer Windows only crashed programs?

Why does the Eurostar not show youth pricing?

Why were contact sensors put on three of the Lunar Module's four legs? Did they ever bend and stick out sideways?

What are the cons of stateless password generators?

Argand formula and more for quaternions?

Interpolating Rotation QuaternionsQuaternions and spatial translationsA question about hypercomplex numbers: quaternions, octonions etcCan different quaternions represent the same orientation?Formula for quaternion exponentiationFormula For N-Dimensional QuaternionsQuaternions. Problem with understanding essence of.How to calculate quaternions from principal axes of an ellipsoid?How to convert Euler angles to Quaternions and get the same Euler angles back from Quaternions?Quaternions statistics from IMU

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0;
}

5

$begingroup$

Is it possible to define a similar form of Argand's formula but for quaternions?

In the sense

$$ cos(nA)+icos(nB)+jcos(nC)+kcos(n) =(cos(A)+icos(B)+jcos(C)+kcos(D))^{n}, $$

where $A, B, C, D$ are the angles of the quaternion with respect the axes $x,y,z,t.$

Also for a quaternion why can you not define the 'numbers'

$ frac{i+j}{k} $ or $ i^{j+k} $ ... where the quaternion is defined in 4-dimensions as

$ a+ib+cj+dk =z? $

quaternions

share|cite|improve this question

edited 8 hours ago

Adrian Keister

6,36277 gold badges2222 silver badges3333 bronze badges

asked 8 hours ago

Jose GarciaJose Garcia

4,28322 gold badges1818 silver badges3939 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you missing a $D$ on the LHS?
  $endgroup$
  – Adrian Keister
  8 hours ago
  

  
  
  
  

add a comment | 

5

$begingroup$

Is it possible to define a similar form of Argand's formula but for quaternions?

In the sense

$$ cos(nA)+icos(nB)+jcos(nC)+kcos(n) =(cos(A)+icos(B)+jcos(C)+kcos(D))^{n}, $$

where $A, B, C, D$ are the angles of the quaternion with respect the axes $x,y,z,t.$

Also for a quaternion why can you not define the 'numbers'

$ frac{i+j}{k} $ or $ i^{j+k} $ ... where the quaternion is defined in 4-dimensions as

$ a+ib+cj+dk =z? $

quaternions

share|cite|improve this question

edited 8 hours ago

Adrian Keister

6,36277 gold badges2222 silver badges3333 bronze badges

asked 8 hours ago

Jose GarciaJose Garcia

4,28322 gold badges1818 silver badges3939 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you missing a $D$ on the LHS?
  $endgroup$
  – Adrian Keister
  8 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Is it possible to define a similar form of Argand's formula but for quaternions?

In the sense

$$ cos(nA)+icos(nB)+jcos(nC)+kcos(n) =(cos(A)+icos(B)+jcos(C)+kcos(D))^{n}, $$

where $A, B, C, D$ are the angles of the quaternion with respect the axes $x,y,z,t.$

Also for a quaternion why can you not define the 'numbers'

$ frac{i+j}{k} $ or $ i^{j+k} $ ... where the quaternion is defined in 4-dimensions as

$ a+ib+cj+dk =z? $

quaternions

share|cite|improve this question

edited 8 hours ago

Adrian Keister

6,36277 gold badges2222 silver badges3333 bronze badges

asked 8 hours ago

Jose GarciaJose Garcia

4,28322 gold badges1818 silver badges3939 bronze badges

$endgroup$

share|cite|improve this question

edited 8 hours ago

Adrian Keister

6,36277 gold badges2222 silver badges3333 bronze badges

asked 8 hours ago

Jose GarciaJose Garcia

4,28322 gold badges1818 silver badges3939 bronze badges

share|cite|improve this question

edited 8 hours ago

Adrian Keister

6,36277 gold badges2222 silver badges3333 bronze badges

asked 8 hours ago

Jose GarciaJose Garcia

4,28322 gold badges1818 silver badges3939 bronze badges

edited 8 hours ago

Adrian Keister

6,36277 gold badges2222 silver badges3333 bronze badges

edited 8 hours ago

Adrian Keister

6,36277 gold badges2222 silver badges3333 bronze badges

asked 8 hours ago

Jose GarciaJose Garcia

4,28322 gold badges1818 silver badges3939 bronze badges

asked 8 hours ago

Jose GarciaJose Garcia

4,28322 gold badges1818 silver badges3939 bronze badges

2 Answers
2

active

oldest

votes

5

$begingroup$

A unit quaternion can be written as
$$q=cos t+(bi+cj+dk)sin t$$
where $b^2+c^2+d^2=1$. Then
$$q^n=cos nt+(bi+cj+dk)sin nt.$$
This just follows from the usual complex case: there's an isomorphism between
$Bbb C$ and the subalgebra generated by $bi+cj+dk$, taking $i$ to
$bi+cj+dk$.

share|cite|improve this answer

answered 8 hours ago

Lord Shark the UnknownLord Shark the Unknown

117k1111 gold badges6767 silver badges148148 bronze badges

$endgroup$

add a comment | 

4

$begingroup$

Since @LordSharktheUnknown discussed the trigonometry, I'll answer your later questions. Do you want $w:=z_1/z_2$ to satisfy $z_1=z_2w$ or $z_1=wz_2$? It matters, which is why we don't usually write such expressions as $frac{i+j}{k}$; you'd want to say $(i+j)k^{-1}$ or $k^{-1}(i+j)$ instead. (These are respectively $-ik-jk=ki-jk=j-i,,i-j$.) That quaternions don't commute also introduces problems with defining exponentiation. Do we want $z_1^{z_2}$ to mean $exp(z_2ln z_1)$ or $exp((ln z_1)z_2)$?

share|cite|improve this answer

answered 8 hours ago

J.G.J.G.

43k22 gold badges3939 silver badges6060 bronze badges

$endgroup$

add a comment | 

Your Answer

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
});


}
});

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3308586%2fargand-formula-and-more-for-quaternions%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

5

$begingroup$

A unit quaternion can be written as
$$q=cos t+(bi+cj+dk)sin t$$
where $b^2+c^2+d^2=1$. Then
$$q^n=cos nt+(bi+cj+dk)sin nt.$$
This just follows from the usual complex case: there's an isomorphism between
$Bbb C$ and the subalgebra generated by $bi+cj+dk$, taking $i$ to
$bi+cj+dk$.

share|cite|improve this answer

answered 8 hours ago

Lord Shark the UnknownLord Shark the Unknown

117k1111 gold badges6767 silver badges148148 bronze badges

$endgroup$

add a comment | 

5

$begingroup$

A unit quaternion can be written as
$$q=cos t+(bi+cj+dk)sin t$$
where $b^2+c^2+d^2=1$. Then
$$q^n=cos nt+(bi+cj+dk)sin nt.$$
This just follows from the usual complex case: there's an isomorphism between
$Bbb C$ and the subalgebra generated by $bi+cj+dk$, taking $i$ to
$bi+cj+dk$.

share|cite|improve this answer

answered 8 hours ago

Lord Shark the UnknownLord Shark the Unknown

117k1111 gold badges6767 silver badges148148 bronze badges

$endgroup$

add a comment | 

$begingroup$

A unit quaternion can be written as
$$q=cos t+(bi+cj+dk)sin t$$
where $b^2+c^2+d^2=1$. Then
$$q^n=cos nt+(bi+cj+dk)sin nt.$$
This just follows from the usual complex case: there's an isomorphism between
$Bbb C$ and the subalgebra generated by $bi+cj+dk$, taking $i$ to
$bi+cj+dk$.

share|cite|improve this answer

answered 8 hours ago

Lord Shark the UnknownLord Shark the Unknown

117k1111 gold badges6767 silver badges148148 bronze badges

$endgroup$

share|cite|improve this answer

answered 8 hours ago

Lord Shark the UnknownLord Shark the Unknown

117k1111 gold badges6767 silver badges148148 bronze badges

answered 8 hours ago

Lord Shark the UnknownLord Shark the Unknown

117k1111 gold badges6767 silver badges148148 bronze badges

answered 8 hours ago

Lord Shark the UnknownLord Shark the Unknown

117k1111 gold badges6767 silver badges148148 bronze badges

4

$begingroup$

Since @LordSharktheUnknown discussed the trigonometry, I'll answer your later questions. Do you want $w:=z_1/z_2$ to satisfy $z_1=z_2w$ or $z_1=wz_2$? It matters, which is why we don't usually write such expressions as $frac{i+j}{k}$; you'd want to say $(i+j)k^{-1}$ or $k^{-1}(i+j)$ instead. (These are respectively $-ik-jk=ki-jk=j-i,,i-j$.) That quaternions don't commute also introduces problems with defining exponentiation. Do we want $z_1^{z_2}$ to mean $exp(z_2ln z_1)$ or $exp((ln z_1)z_2)$?

share|cite|improve this answer

answered 8 hours ago

J.G.J.G.

43k22 gold badges3939 silver badges6060 bronze badges

$endgroup$

add a comment | 

4

$begingroup$

Since @LordSharktheUnknown discussed the trigonometry, I'll answer your later questions. Do you want $w:=z_1/z_2$ to satisfy $z_1=z_2w$ or $z_1=wz_2$? It matters, which is why we don't usually write such expressions as $frac{i+j}{k}$; you'd want to say $(i+j)k^{-1}$ or $k^{-1}(i+j)$ instead. (These are respectively $-ik-jk=ki-jk=j-i,,i-j$.) That quaternions don't commute also introduces problems with defining exponentiation. Do we want $z_1^{z_2}$ to mean $exp(z_2ln z_1)$ or $exp((ln z_1)z_2)$?

share|cite|improve this answer

answered 8 hours ago

J.G.J.G.

43k22 gold badges3939 silver badges6060 bronze badges

$endgroup$

add a comment | 

$begingroup$

Since @LordSharktheUnknown discussed the trigonometry, I'll answer your later questions. Do you want $w:=z_1/z_2$ to satisfy $z_1=z_2w$ or $z_1=wz_2$? It matters, which is why we don't usually write such expressions as $frac{i+j}{k}$; you'd want to say $(i+j)k^{-1}$ or $k^{-1}(i+j)$ instead. (These are respectively $-ik-jk=ki-jk=j-i,,i-j$.) That quaternions don't commute also introduces problems with defining exponentiation. Do we want $z_1^{z_2}$ to mean $exp(z_2ln z_1)$ or $exp((ln z_1)z_2)$?

share|cite|improve this answer

answered 8 hours ago

J.G.J.G.

43k22 gold badges3939 silver badges6060 bronze badges

$endgroup$

share|cite|improve this answer

answered 8 hours ago

J.G.J.G.

43k22 gold badges3939 silver badges6060 bronze badges

answered 8 hours ago

J.G.J.G.

43k22 gold badges3939 silver badges6060 bronze badges

answered 8 hours ago

J.G.J.G.

43k22 gold badges3939 silver badges6060 bronze badges