Atypical way to find angle between unit vectors: $theta = 2 sin^-1left(frac12left|hatA-hatBright|right)$
Clash Royale CLAN TAG#URR8PPP
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At my work, I have come across code with the following way of calculating the angle between two vectors.
$$theta = 2 sin^-1left(frac12left|hatA-hatB right|right)$$
(Note the physics convention: $hatv$ indicates the normalization of $v$; ie, $hatv:=v/|v|$).
I've spent some time, but I can't think of how this was derived using typical methodologies (law of sines, law of cosines, dot product, cross product). It is pretty different. So my questions is,
How could this have been derived?
linear-algebra trigonometry vectors
edited Jan 22 at 16:42
Blue
48.4k870154
asked Jan 21 at 21:49
WolcottRWolcottR
455
$endgroup$
add a comment |
$begingroup$
At my work, I have come across code with the following way of calculating the angle between two vectors.
$$theta = 2 sin^-1left(frac12left|hatA-hatB right|right)$$
(Note the physics convention: $hatv$ indicates the normalization of $v$; ie, $hatv:=v/|v|$).
I've spent some time, but I can't think of how this was derived using typical methodologies (law of sines, law of cosines, dot product, cross product). It is pretty different. So my questions is,
How could this have been derived?
linear-algebra trigonometry vectors
edited Jan 22 at 16:42
Blue
48.4k870154
asked Jan 21 at 21:49
WolcottRWolcottR
455
$endgroup$
$begingroup$
@DavidK Thank you. I have addressed that.
$endgroup$
– WolcottR
Jan 22 at 15:46
add a comment |
$begingroup$
At my work, I have come across code with the following way of calculating the angle between two vectors.
$$theta = 2 sin^-1left(frac12left|hatA-hatB right|right)$$
(Note the physics convention: $hatv$ indicates the normalization of $v$; ie, $hatv:=v/|v|$).
I've spent some time, but I can't think of how this was derived using typical methodologies (law of sines, law of cosines, dot product, cross product). It is pretty different. So my questions is,
How could this have been derived?
linear-algebra trigonometry vectors
edited Jan 22 at 16:42
Blue
48.4k870154
asked Jan 21 at 21:49
WolcottRWolcottR
455
$endgroup$
At my work, I have come across code with the following way of calculating the angle between two vectors.
$$theta = 2 sin^-1left(frac12left|hatA-hatB right|right)$$
(Note the physics convention: $hatv$ indicates the normalization of $v$; ie, $hatv:=v/|v|$).
I've spent some time, but I can't think of how this was derived using typical methodologies (law of sines, law of cosines, dot product, cross product). It is pretty different. So my questions is,
How could this have been derived?
linear-algebra trigonometry vectors
linear-algebra trigonometry vectors
edited Jan 22 at 16:42
Blue
48.4k870154
asked Jan 21 at 21:49
WolcottRWolcottR
455
edited Jan 22 at 16:42
Blue
48.4k870154
asked Jan 21 at 21:49
WolcottRWolcottR
455
edited Jan 22 at 16:42
Blue
48.4k870154
edited Jan 22 at 16:42
Blue
48.4k870154
edited Jan 22 at 16:42
Blue
48.4k870154
48.4k870154
asked Jan 21 at 21:49
WolcottRWolcottR
455
asked Jan 21 at 21:49
WolcottRWolcottR
455
asked Jan 21 at 21:49
WolcottRWolcottR
455
455
$begingroup$
@DavidK Thank you. I have addressed that.
$endgroup$
– WolcottR
Jan 22 at 15:46
add a comment |
$begingroup$
@DavidK Thank you. I have addressed that.
$endgroup$
– WolcottR
Jan 22 at 15:46
$begingroup$
@DavidK Thank you. I have addressed that.
$endgroup$
– WolcottR
Jan 22 at 15:46
$begingroup$
@DavidK Thank you. I have addressed that.
$endgroup$
– WolcottR
Jan 22 at 15:46
add a comment |
1 Answer
1
active
oldest
votes
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This works when $A$ and $B$ are unit vectors.
You have
$$
|A-B|^2=(A-B)cdot(A-B)=|A|^2+|B|^2-2,Acdot B=2-2costheta=4sin^2tfractheta2.
$$
Solving,
$$
theta=2arcsintfrac2.
$$
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
$endgroup$
1
$begingroup$
A general comment: note the hats on the vectors, which often indicates normalization.
$endgroup$
– Will R
Jan 21 at 21:56
1
$begingroup$
Good to know! I haven't seen that in the last 30 years of doing math.
$endgroup$
– Martin Argerami
Jan 21 at 21:57
1
$begingroup$
It's fairly common in physics, I think. Some people denote the standard unit vectors in $mathbbR^3$ with $i,j,k,$ and some of those people put hats on them, to emphasize that they are unit vectors.
$endgroup$
– Will R
Jan 21 at 22:02
3
$begingroup$
I have been using the hats on $i,j,k$ for many years, but I never saw the notation being used for other vectors.
$endgroup$
– Martin Argerami
Jan 21 at 22:16
2
$begingroup$
@WillR It's indeed common in physics. We often go one step further too, and write stuff like $vecv=vhatv$. Makes for fairly compact notation.
$endgroup$
– Anyon
Jan 22 at 2:47
|
show 1 more comment
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This works when $A$ and $B$ are unit vectors.
You have
$$
|A-B|^2=(A-B)cdot(A-B)=|A|^2+|B|^2-2,Acdot B=2-2costheta=4sin^2tfractheta2.
$$
Solving,
$$
theta=2arcsintfrac2.
$$
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
$endgroup$
1
$begingroup$
A general comment: note the hats on the vectors, which often indicates normalization.
$endgroup$
– Will R
Jan 21 at 21:56
1
$begingroup$
Good to know! I haven't seen that in the last 30 years of doing math.
$endgroup$
– Martin Argerami
Jan 21 at 21:57
1
$begingroup$
It's fairly common in physics, I think. Some people denote the standard unit vectors in $mathbbR^3$ with $i,j,k,$ and some of those people put hats on them, to emphasize that they are unit vectors.
$endgroup$
– Will R
Jan 21 at 22:02
3
$begingroup$
I have been using the hats on $i,j,k$ for many years, but I never saw the notation being used for other vectors.
$endgroup$
– Martin Argerami
Jan 21 at 22:16
2
$begingroup$
@WillR It's indeed common in physics. We often go one step further too, and write stuff like $vecv=vhatv$. Makes for fairly compact notation.
$endgroup$
– Anyon
Jan 22 at 2:47
|
show 1 more comment
$begingroup$
This works when $A$ and $B$ are unit vectors.
You have
$$
|A-B|^2=(A-B)cdot(A-B)=|A|^2+|B|^2-2,Acdot B=2-2costheta=4sin^2tfractheta2.
$$
Solving,
$$
theta=2arcsintfrac2.
$$
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
$endgroup$
1
$begingroup$
A general comment: note the hats on the vectors, which often indicates normalization.
$endgroup$
– Will R
Jan 21 at 21:56
1
$begingroup$
Good to know! I haven't seen that in the last 30 years of doing math.
$endgroup$
– Martin Argerami
Jan 21 at 21:57
1
$begingroup$
It's fairly common in physics, I think. Some people denote the standard unit vectors in $mathbbR^3$ with $i,j,k,$ and some of those people put hats on them, to emphasize that they are unit vectors.
$endgroup$
– Will R
Jan 21 at 22:02
3
$begingroup$
I have been using the hats on $i,j,k$ for many years, but I never saw the notation being used for other vectors.
$endgroup$
– Martin Argerami
Jan 21 at 22:16
2
$begingroup$
@WillR It's indeed common in physics. We often go one step further too, and write stuff like $vecv=vhatv$. Makes for fairly compact notation.
$endgroup$
– Anyon
Jan 22 at 2:47
|
show 1 more comment
$begingroup$
This works when $A$ and $B$ are unit vectors.
You have
$$
|A-B|^2=(A-B)cdot(A-B)=|A|^2+|B|^2-2,Acdot B=2-2costheta=4sin^2tfractheta2.
$$
Solving,
$$
theta=2arcsintfrac2.
$$
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
$endgroup$
This works when $A$ and $B$ are unit vectors.
You have
$$
|A-B|^2=(A-B)cdot(A-B)=|A|^2+|B|^2-2,Acdot B=2-2costheta=4sin^2tfractheta2.
$$
Solving,
$$
theta=2arcsintfrac2.
$$
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
answered Jan 21 at 21:54
Martin ArgeramiMartin Argerami
127k1182181
127k1182181
1
$begingroup$
A general comment: note the hats on the vectors, which often indicates normalization.
$endgroup$
– Will R
Jan 21 at 21:56
1
$begingroup$
Good to know! I haven't seen that in the last 30 years of doing math.
$endgroup$
– Martin Argerami
Jan 21 at 21:57
1
$begingroup$
It's fairly common in physics, I think. Some people denote the standard unit vectors in $mathbbR^3$ with $i,j,k,$ and some of those people put hats on them, to emphasize that they are unit vectors.
$endgroup$
– Will R
Jan 21 at 22:02
3
$begingroup$
I have been using the hats on $i,j,k$ for many years, but I never saw the notation being used for other vectors.
$endgroup$
– Martin Argerami
Jan 21 at 22:16
2
$begingroup$
@WillR It's indeed common in physics. We often go one step further too, and write stuff like $vecv=vhatv$. Makes for fairly compact notation.
$endgroup$
– Anyon
Jan 22 at 2:47
|
show 1 more comment
1
$begingroup$
A general comment: note the hats on the vectors, which often indicates normalization.
$endgroup$
– Will R
Jan 21 at 21:56
1
$begingroup$
Good to know! I haven't seen that in the last 30 years of doing math.
$endgroup$
– Martin Argerami
Jan 21 at 21:57
1
$begingroup$
It's fairly common in physics, I think. Some people denote the standard unit vectors in $mathbbR^3$ with $i,j,k,$ and some of those people put hats on them, to emphasize that they are unit vectors.
$endgroup$
– Will R
Jan 21 at 22:02
3
$begingroup$
I have been using the hats on $i,j,k$ for many years, but I never saw the notation being used for other vectors.
$endgroup$
– Martin Argerami
Jan 21 at 22:16
2
$begingroup$
@WillR It's indeed common in physics. We often go one step further too, and write stuff like $vecv=vhatv$. Makes for fairly compact notation.
$endgroup$
– Anyon
Jan 22 at 2:47
1
1
$begingroup$
A general comment: note the hats on the vectors, which often indicates normalization.
$endgroup$
– Will R
Jan 21 at 21:56
$begingroup$
A general comment: note the hats on the vectors, which often indicates normalization.
$endgroup$
– Will R
Jan 21 at 21:56
1
1
$begingroup$
Good to know! I haven't seen that in the last 30 years of doing math.
$endgroup$
– Martin Argerami
Jan 21 at 21:57
$begingroup$
Good to know! I haven't seen that in the last 30 years of doing math.
$endgroup$
– Martin Argerami
Jan 21 at 21:57
1
1
$begingroup$
It's fairly common in physics, I think. Some people denote the standard unit vectors in $mathbbR^3$ with $i,j,k,$ and some of those people put hats on them, to emphasize that they are unit vectors.
$endgroup$
– Will R
Jan 21 at 22:02
$begingroup$
It's fairly common in physics, I think. Some people denote the standard unit vectors in $mathbbR^3$ with $i,j,k,$ and some of those people put hats on them, to emphasize that they are unit vectors.
$endgroup$
– Will R
Jan 21 at 22:02
3
3
$begingroup$
I have been using the hats on $i,j,k$ for many years, but I never saw the notation being used for other vectors.
$endgroup$
– Martin Argerami
Jan 21 at 22:16
$begingroup$
I have been using the hats on $i,j,k$ for many years, but I never saw the notation being used for other vectors.
$endgroup$
– Martin Argerami
Jan 21 at 22:16
2
2
$begingroup$
@WillR It's indeed common in physics. We often go one step further too, and write stuff like $vecv=vhatv$. Makes for fairly compact notation.
$endgroup$
– Anyon
Jan 22 at 2:47
$begingroup$
@WillR It's indeed common in physics. We often go one step further too, and write stuff like $vecv=vhatv$. Makes for fairly compact notation.
$endgroup$
– Anyon
Jan 22 at 2:47
|
show 1 more comment
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$begingroup$
@DavidK Thank you. I have addressed that.
$endgroup$
– WolcottR
Jan 22 at 15:46