Find the minimum distance from the point a to the set $X={ x=(x_1,x_2,…x_n)mid b_1x_1+b_2x_2+…+b_nx_n=c...
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Find the minimum distance from the point $a=(a_1,a_2,...a_n)$ to the set $$X={ x=(x_1,x_2,...x_n)mid b_1x_1+b_2x_2+...+b_nx_n=c }$$
where $ b_1^2 + b_2^2 +...+ b_n^2 gt 0 $ and $b_1, b_2,...,b_n,c in Bbb R$.
I have found so far the definition of distance from a point to a set in different books where stands that if (X,d) is a metric space $Esubset X$, $Eneq varnothing$ and $xin X$ we can define the distance from the point x to the set E in the following way
$$d(x,E):=Inf{d(x,y): y in E}$$
but I have not found any example that could help me solve this problem, and I have no Idea of how to approach it
linear-algebra multivariable-calculus
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
$endgroup$
add a comment |
$begingroup$
Find the minimum distance from the point $a=(a_1,a_2,...a_n)$ to the set $$X={ x=(x_1,x_2,...x_n)mid b_1x_1+b_2x_2+...+b_nx_n=c }$$
where $ b_1^2 + b_2^2 +...+ b_n^2 gt 0 $ and $b_1, b_2,...,b_n,c in Bbb R$.
I have found so far the definition of distance from a point to a set in different books where stands that if (X,d) is a metric space $Esubset X$, $Eneq varnothing$ and $xin X$ we can define the distance from the point x to the set E in the following way
$$d(x,E):=Inf{d(x,y): y in E}$$
but I have not found any example that could help me solve this problem, and I have no Idea of how to approach it
linear-algebra multivariable-calculus
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
$endgroup$
add a comment |
$begingroup$
Find the minimum distance from the point $a=(a_1,a_2,...a_n)$ to the set $$X={ x=(x_1,x_2,...x_n)mid b_1x_1+b_2x_2+...+b_nx_n=c }$$
where $ b_1^2 + b_2^2 +...+ b_n^2 gt 0 $ and $b_1, b_2,...,b_n,c in Bbb R$.
I have found so far the definition of distance from a point to a set in different books where stands that if (X,d) is a metric space $Esubset X$, $Eneq varnothing$ and $xin X$ we can define the distance from the point x to the set E in the following way
$$d(x,E):=Inf{d(x,y): y in E}$$
but I have not found any example that could help me solve this problem, and I have no Idea of how to approach it
linear-algebra multivariable-calculus
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
$endgroup$
Find the minimum distance from the point $a=(a_1,a_2,...a_n)$ to the set $$X={ x=(x_1,x_2,...x_n)mid b_1x_1+b_2x_2+...+b_nx_n=c }$$
where $ b_1^2 + b_2^2 +...+ b_n^2 gt 0 $ and $b_1, b_2,...,b_n,c in Bbb R$.
I have found so far the definition of distance from a point to a set in different books where stands that if (X,d) is a metric space $Esubset X$, $Eneq varnothing$ and $xin X$ we can define the distance from the point x to the set E in the following way
$$d(x,E):=Inf{d(x,y): y in E}$$
but I have not found any example that could help me solve this problem, and I have no Idea of how to approach it
linear-algebra multivariable-calculus
linear-algebra multivariable-calculus
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
edited Jan 14 at 7:09
Iván Galeana Aguilar
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
asked Jan 14 at 7:03
Iván Galeana AguilarIván Galeana Aguilar
1389
1389
add a comment |
add a comment |
1 Answer
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The method of Lagrange Multipliers tells you how to find the minimum value of $sumlimits_{k=1}^{n} (x_i-a_i)^{2}$ subject to the condition $sumlimits_{k=1}^{n} b_ix_i=c$. Consider the function $sumlimits_{k=1}^{n} (x_i-a_i)^{2} -lambda (sumlimits_{k=1}^{n} b_ix_i-c)$ where $lambda$ is a new parameter. The point where the minimum occurs can be found by setting derivatives w.r.t $x_1,x_2,cdots,x_n,lambda$ equal to $0$. This gives $2(x_k-a_k)-lambda b_k=0$ for eaxh $k leq n$. Solve this for $x_k$ in terms of $lambda$ and then use the given condition $sumlimits_{k=1}^{n} b_ix_i=c$ to find the value of $lambda$.
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
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1 Answer
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1 Answer
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$begingroup$
The method of Lagrange Multipliers tells you how to find the minimum value of $sumlimits_{k=1}^{n} (x_i-a_i)^{2}$ subject to the condition $sumlimits_{k=1}^{n} b_ix_i=c$. Consider the function $sumlimits_{k=1}^{n} (x_i-a_i)^{2} -lambda (sumlimits_{k=1}^{n} b_ix_i-c)$ where $lambda$ is a new parameter. The point where the minimum occurs can be found by setting derivatives w.r.t $x_1,x_2,cdots,x_n,lambda$ equal to $0$. This gives $2(x_k-a_k)-lambda b_k=0$ for eaxh $k leq n$. Solve this for $x_k$ in terms of $lambda$ and then use the given condition $sumlimits_{k=1}^{n} b_ix_i=c$ to find the value of $lambda$.
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
$begingroup$
The method of Lagrange Multipliers tells you how to find the minimum value of $sumlimits_{k=1}^{n} (x_i-a_i)^{2}$ subject to the condition $sumlimits_{k=1}^{n} b_ix_i=c$. Consider the function $sumlimits_{k=1}^{n} (x_i-a_i)^{2} -lambda (sumlimits_{k=1}^{n} b_ix_i-c)$ where $lambda$ is a new parameter. The point where the minimum occurs can be found by setting derivatives w.r.t $x_1,x_2,cdots,x_n,lambda$ equal to $0$. This gives $2(x_k-a_k)-lambda b_k=0$ for eaxh $k leq n$. Solve this for $x_k$ in terms of $lambda$ and then use the given condition $sumlimits_{k=1}^{n} b_ix_i=c$ to find the value of $lambda$.
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
$begingroup$
The method of Lagrange Multipliers tells you how to find the minimum value of $sumlimits_{k=1}^{n} (x_i-a_i)^{2}$ subject to the condition $sumlimits_{k=1}^{n} b_ix_i=c$. Consider the function $sumlimits_{k=1}^{n} (x_i-a_i)^{2} -lambda (sumlimits_{k=1}^{n} b_ix_i-c)$ where $lambda$ is a new parameter. The point where the minimum occurs can be found by setting derivatives w.r.t $x_1,x_2,cdots,x_n,lambda$ equal to $0$. This gives $2(x_k-a_k)-lambda b_k=0$ for eaxh $k leq n$. Solve this for $x_k$ in terms of $lambda$ and then use the given condition $sumlimits_{k=1}^{n} b_ix_i=c$ to find the value of $lambda$.
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
The method of Lagrange Multipliers tells you how to find the minimum value of $sumlimits_{k=1}^{n} (x_i-a_i)^{2}$ subject to the condition $sumlimits_{k=1}^{n} b_ix_i=c$. Consider the function $sumlimits_{k=1}^{n} (x_i-a_i)^{2} -lambda (sumlimits_{k=1}^{n} b_ix_i-c)$ where $lambda$ is a new parameter. The point where the minimum occurs can be found by setting derivatives w.r.t $x_1,x_2,cdots,x_n,lambda$ equal to $0$. This gives $2(x_k-a_k)-lambda b_k=0$ for eaxh $k leq n$. Solve this for $x_k$ in terms of $lambda$ and then use the given condition $sumlimits_{k=1}^{n} b_ix_i=c$ to find the value of $lambda$.
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Jan 14 at 7:20
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
69.1k53169
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