Mixing
Angle between sum of vectors
$begingroup$
Let $u,v$ and $w$ be vectors in $mathbb{R}^n$ and let $theta(u,w), theta(v,w)$ and $theta(u+v,w)$ represent the angle between each listed pair of vectors. Does it hold that one of the following two statements must be true:
$$theta(u,w) geq theta(u+v,w)$$ Or:
$$theta(v,w) geq theta(u+v,w)$$
I feel this must be true and that it follows from some simple property that I'm forgetting. If it is true, does it generalize to any finite list of vectors?
geometry vectors
asked Jan 18 at 15:48
ghiufheghiufhe
584
$endgroup$
add a comment |
$begingroup$
Let $u,v$ and $w$ be vectors in $mathbb{R}^n$ and let $theta(u,w), theta(v,w)$ and $theta(u+v,w)$ represent the angle between each listed pair of vectors. Does it hold that one of the following two statements must be true:
$$theta(u,w) geq theta(u+v,w)$$ Or:
$$theta(v,w) geq theta(u+v,w)$$
I feel this must be true and that it follows from some simple property that I'm forgetting. If it is true, does it generalize to any finite list of vectors?
geometry vectors
asked Jan 18 at 15:48
ghiufheghiufhe
584
$endgroup$
add a comment |
$begingroup$
Let $u,v$ and $w$ be vectors in $mathbb{R}^n$ and let $theta(u,w), theta(v,w)$ and $theta(u+v,w)$ represent the angle between each listed pair of vectors. Does it hold that one of the following two statements must be true:
$$theta(u,w) geq theta(u+v,w)$$ Or:
$$theta(v,w) geq theta(u+v,w)$$
I feel this must be true and that it follows from some simple property that I'm forgetting. If it is true, does it generalize to any finite list of vectors?
geometry vectors
asked Jan 18 at 15:48
ghiufheghiufhe
584
$endgroup$
Let $u,v$ and $w$ be vectors in $mathbb{R}^n$ and let $theta(u,w), theta(v,w)$ and $theta(u+v,w)$ represent the angle between each listed pair of vectors. Does it hold that one of the following two statements must be true:
$$theta(u,w) geq theta(u+v,w)$$ Or:
$$theta(v,w) geq theta(u+v,w)$$
I feel this must be true and that it follows from some simple property that I'm forgetting. If it is true, does it generalize to any finite list of vectors?
geometry vectors
geometry vectors
asked Jan 18 at 15:48
ghiufheghiufhe
584
asked Jan 18 at 15:48
ghiufheghiufhe
584
asked Jan 18 at 15:48
ghiufheghiufhe
584
asked Jan 18 at 15:48
ghiufheghiufhe
584
asked Jan 18 at 15:48
ghiufheghiufhe
584
584
add a comment |
add a comment |
1 Answer
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I don't think this is true. Take the vectors $u=langle-1, 1rangle$, $v=langle 1, 1rangle$ and $w=langle 0, -1rangle$ in $mathbb{R}^2$. Then $u+v=langle 0, 2rangle$, and $theta(u+v, w)=pi$, while both $theta(u,w)$ and $theta(v,w)=frac{3pi}{4}$.
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
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1 Answer
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1 Answer
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$begingroup$
I don't think this is true. Take the vectors $u=langle-1, 1rangle$, $v=langle 1, 1rangle$ and $w=langle 0, -1rangle$ in $mathbb{R}^2$. Then $u+v=langle 0, 2rangle$, and $theta(u+v, w)=pi$, while both $theta(u,w)$ and $theta(v,w)=frac{3pi}{4}$.
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
$endgroup$
add a comment |
$begingroup$
I don't think this is true. Take the vectors $u=langle-1, 1rangle$, $v=langle 1, 1rangle$ and $w=langle 0, -1rangle$ in $mathbb{R}^2$. Then $u+v=langle 0, 2rangle$, and $theta(u+v, w)=pi$, while both $theta(u,w)$ and $theta(v,w)=frac{3pi}{4}$.
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
$endgroup$
add a comment |
$begingroup$
I don't think this is true. Take the vectors $u=langle-1, 1rangle$, $v=langle 1, 1rangle$ and $w=langle 0, -1rangle$ in $mathbb{R}^2$. Then $u+v=langle 0, 2rangle$, and $theta(u+v, w)=pi$, while both $theta(u,w)$ and $theta(v,w)=frac{3pi}{4}$.
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
$endgroup$
I don't think this is true. Take the vectors $u=langle-1, 1rangle$, $v=langle 1, 1rangle$ and $w=langle 0, -1rangle$ in $mathbb{R}^2$. Then $u+v=langle 0, 2rangle$, and $theta(u+v, w)=pi$, while both $theta(u,w)$ and $theta(v,w)=frac{3pi}{4}$.
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
answered Jan 18 at 16:09
Calvin GodfreyCalvin Godfrey
633311
633311
add a comment |
add a comment |
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