How does “$t$” disappear when finding the distance from a point to a line?

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I am trying to see why the "t" disappears when finding the distance from a point to a line in the explanation on wikipedia under the section called Vector Formulation on this page:



https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line



On that page you can see $x= a+ tn$ is the vector, and $p$ is the point. But then somehow the $t$ drops out. In particular, I do not understand this sentence on that page:




Then $(a -p )cdot n)n,$ is the projected length onto the line...




I do not understand how we got that expression, and the $t$ dropped out somehow.




vector-spaces vectors projection




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    I am trying to see why the "t" disappears when finding the distance from a point to a line in the explanation on wikipedia under the section called Vector Formulation on this page:



    https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line



    On that page you can see $x= a+ tn$ is the vector, and $p$ is the point. But then somehow the $t$ drops out. In particular, I do not understand this sentence on that page:




    Then $(a -p )cdot n)n,$ is the projected length onto the line...




    I do not understand how we got that expression, and the $t$ dropped out somehow.




    vector-spaces vectors projection




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I am trying to see why the "t" disappears when finding the distance from a point to a line in the explanation on wikipedia under the section called Vector Formulation on this page:



      https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line



      On that page you can see $x= a+ tn$ is the vector, and $p$ is the point. But then somehow the $t$ drops out. In particular, I do not understand this sentence on that page:




      Then $(a -p )cdot n)n,$ is the projected length onto the line...




      I do not understand how we got that expression, and the $t$ dropped out somehow.




      vector-spaces vectors projection




      share|cite|improve this question











      I am trying to see why the "t" disappears when finding the distance from a point to a line in the explanation on wikipedia under the section called Vector Formulation on this page:



      https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line



      On that page you can see $x= a+ tn$ is the vector, and $p$ is the point. But then somehow the $t$ drops out. In particular, I do not understand this sentence on that page:




      Then $(a -p )cdot n)n,$ is the projected length onto the line...




      I do not understand how we got that expression, and the $t$ dropped out somehow.





      vector-spaces vectors projection





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      share|cite|improve this question




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      asked Jul 28 at 22:34









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          1. $a-p$ is another notation for the translation vector which moves $p$ to $a$, in other words the vector $overrightarrowpa$.

          2. You can easily check the vector $; a-p -langle a-p,nrangle n$ is orthogonal to $n$,hence its norm is the distance from point $p$ to the line.





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            1. $a-p$ is another notation for the translation vector which moves $p$ to $a$, in other words the vector $overrightarrowpa$.

            2. You can easily check the vector $; a-p -langle a-p,nrangle n$ is orthogonal to $n$,hence its norm is the distance from point $p$ to the line.





            share|cite|improve this answer

























              up vote
              0
              down vote













              1. $a-p$ is another notation for the translation vector which moves $p$ to $a$, in other words the vector $overrightarrowpa$.

              2. You can easily check the vector $; a-p -langle a-p,nrangle n$ is orthogonal to $n$,hence its norm is the distance from point $p$ to the line.





              share|cite|improve this answer























                up vote
                0
                down vote










                up vote
                0
                down vote









                1. $a-p$ is another notation for the translation vector which moves $p$ to $a$, in other words the vector $overrightarrowpa$.

                2. You can easily check the vector $; a-p -langle a-p,nrangle n$ is orthogonal to $n$,hence its norm is the distance from point $p$ to the line.





                share|cite|improve this answer













                1. $a-p$ is another notation for the translation vector which moves $p$ to $a$, in other words the vector $overrightarrowpa$.

                2. You can easily check the vector $; a-p -langle a-p,nrangle n$ is orthogonal to $n$,hence its norm is the distance from point $p$ to the line.






                share|cite|improve this answer













                share|cite|improve this answer



                share|cite|improve this answer











                answered Jul 28 at 22:46









                Bernard

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