Mixing
Projection from point onto plane
$begingroup$
Let the plane is $prod:vec{x}bullethat{n}=d$ be a plane, where $vec{x}in R^{3}$ and $din R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the plane is equal
$$Proj_{prod}(vec{x})=vec{x}+(d-vec{x}bullethat{n})hat{n}.$$
Let $vec{z}$ be any point on the plane then $vec{z}bullethat{n}=d$, now the distance between $vec{z}$ and any point $vec{x}in R^3$ is
$$||vec{z}-vec{x}||$$ and
$$Rightarrow ~big(||vec{z}-vec{x}||big)^2=big(||hat{n}||big)^2 big(||vec{z}-vec{x}||big)^2 =big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2$$,where ||hat{n}||=1.
I have problem in the above equality, I know that the Cauchy-Schwarz inequality is $$|<vec{x},vec{y}>|leq||vec{x}||||vec{y}||$$. How can I use this Cauchy-Schwarz inequality in the above case that
$$big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2geq|hat{n}bullet(vec{z}-vec{x})|^2$$
The equality holds only $iff$ $hat{n}times(vec{y}-vec{x})=0$
I can proceed further to get the desired projection.
geometry functional-analysis vector-analysis isometry
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
$endgroup$
add a comment |
$begingroup$
Let the plane is $prod:vec{x}bullethat{n}=d$ be a plane, where $vec{x}in R^{3}$ and $din R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the plane is equal
$$Proj_{prod}(vec{x})=vec{x}+(d-vec{x}bullethat{n})hat{n}.$$
Let $vec{z}$ be any point on the plane then $vec{z}bullethat{n}=d$, now the distance between $vec{z}$ and any point $vec{x}in R^3$ is
$$||vec{z}-vec{x}||$$ and
$$Rightarrow ~big(||vec{z}-vec{x}||big)^2=big(||hat{n}||big)^2 big(||vec{z}-vec{x}||big)^2 =big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2$$,where ||hat{n}||=1.
I have problem in the above equality, I know that the Cauchy-Schwarz inequality is $$|<vec{x},vec{y}>|leq||vec{x}||||vec{y}||$$. How can I use this Cauchy-Schwarz inequality in the above case that
$$big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2geq|hat{n}bullet(vec{z}-vec{x})|^2$$
The equality holds only $iff$ $hat{n}times(vec{y}-vec{x})=0$
I can proceed further to get the desired projection.
geometry functional-analysis vector-analysis isometry
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
$endgroup$
$begingroup$
I’m sure that whoever gave you this assignment would also like to know if you can show this. Have you tried anything at all? Where are you running into difficulty?
$endgroup$
– amd
Jan 17 at 17:55
$begingroup$
@amd Sir I explain what I have done and where I am getting problem. Please check if you can help!
$endgroup$
– Noor Aslam
Jan 17 at 18:22
add a comment |
$begingroup$
Let the plane is $prod:vec{x}bullethat{n}=d$ be a plane, where $vec{x}in R^{3}$ and $din R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the plane is equal
$$Proj_{prod}(vec{x})=vec{x}+(d-vec{x}bullethat{n})hat{n}.$$
Let $vec{z}$ be any point on the plane then $vec{z}bullethat{n}=d$, now the distance between $vec{z}$ and any point $vec{x}in R^3$ is
$$||vec{z}-vec{x}||$$ and
$$Rightarrow ~big(||vec{z}-vec{x}||big)^2=big(||hat{n}||big)^2 big(||vec{z}-vec{x}||big)^2 =big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2$$,where ||hat{n}||=1.
I have problem in the above equality, I know that the Cauchy-Schwarz inequality is $$|<vec{x},vec{y}>|leq||vec{x}||||vec{y}||$$. How can I use this Cauchy-Schwarz inequality in the above case that
$$big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2geq|hat{n}bullet(vec{z}-vec{x})|^2$$
The equality holds only $iff$ $hat{n}times(vec{y}-vec{x})=0$
I can proceed further to get the desired projection.
geometry functional-analysis vector-analysis isometry
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
$endgroup$
Let the plane is $prod:vec{x}bullethat{n}=d$ be a plane, where $vec{x}in R^{3}$ and $din R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the plane is equal
$$Proj_{prod}(vec{x})=vec{x}+(d-vec{x}bullethat{n})hat{n}.$$
Let $vec{z}$ be any point on the plane then $vec{z}bullethat{n}=d$, now the distance between $vec{z}$ and any point $vec{x}in R^3$ is
$$||vec{z}-vec{x}||$$ and
$$Rightarrow ~big(||vec{z}-vec{x}||big)^2=big(||hat{n}||big)^2 big(||vec{z}-vec{x}||big)^2 =big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2$$,where ||hat{n}||=1.
I have problem in the above equality, I know that the Cauchy-Schwarz inequality is $$|<vec{x},vec{y}>|leq||vec{x}||||vec{y}||$$. How can I use this Cauchy-Schwarz inequality in the above case that
$$big(||hat{n}times(vec{z}-vec{x})||big)^2+|hat{n}bullet(vec{z}-vec{x})|^2geq|hat{n}bullet(vec{z}-vec{x})|^2$$
The equality holds only $iff$ $hat{n}times(vec{y}-vec{x})=0$
I can proceed further to get the desired projection.
geometry functional-analysis vector-analysis isometry
geometry functional-analysis vector-analysis isometry
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
edited Jan 17 at 18:21
Noor Aslam
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
asked Jan 17 at 17:37
Noor AslamNoor Aslam
14612
14612
$begingroup$
I’m sure that whoever gave you this assignment would also like to know if you can show this. Have you tried anything at all? Where are you running into difficulty?
$endgroup$
– amd
Jan 17 at 17:55
$begingroup$
@amd Sir I explain what I have done and where I am getting problem. Please check if you can help!
$endgroup$
– Noor Aslam
Jan 17 at 18:22
add a comment |
$begingroup$
I’m sure that whoever gave you this assignment would also like to know if you can show this. Have you tried anything at all? Where are you running into difficulty?
$endgroup$
– amd
Jan 17 at 17:55
$begingroup$
@amd Sir I explain what I have done and where I am getting problem. Please check if you can help!
$endgroup$
– Noor Aslam
Jan 17 at 18:22
$begingroup$
I’m sure that whoever gave you this assignment would also like to know if you can show this. Have you tried anything at all? Where are you running into difficulty?
$endgroup$
– amd
Jan 17 at 17:55
$begingroup$
I’m sure that whoever gave you this assignment would also like to know if you can show this. Have you tried anything at all? Where are you running into difficulty?
$endgroup$
– amd
Jan 17 at 17:55
$begingroup$
@amd Sir I explain what I have done and where I am getting problem. Please check if you can help!
$endgroup$
– Noor Aslam
Jan 17 at 18:22
$begingroup$
@amd Sir I explain what I have done and where I am getting problem. Please check if you can help!
$endgroup$
– Noor Aslam
Jan 17 at 18:22
add a comment |
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$begingroup$
I’m sure that whoever gave you this assignment would also like to know if you can show this. Have you tried anything at all? Where are you running into difficulty?
$endgroup$
– amd
Jan 17 at 17:55
$begingroup$
@amd Sir I explain what I have done and where I am getting problem. Please check if you can help!
$endgroup$
– Noor Aslam
Jan 17 at 18:22