Plane with normal vectora) If unit normal vector is ( a 1 , b 1 , c 1 ) , , then, how the point P 1 on the plane becomes ( D a 1 , D b 1 , D c 1 ) ? ?b) If unit normal is ( 1 / 3 , 2 / 3 , 2 / 3 ) then P 1 becomes ( 2 / 3 , 4 / 3 , 4 / 3 ) Where D = 2. We know that normal vector began on the plane at point P 1 and ends at ( 1 / 3 , 2 / 3 , 2 / 3 ) . My questions is how unit normal vector coordinates value less than P 1 coordinates value, because unit normal vector pointing outside of the plane it should be greater coordinates value than P 1 ?

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Plane with normal vectora) If unit normal vector is   (      a    1    ,      b    1    ,      c    1    )  , , then, how the point       P    1   on the plane becomes   (  D      a    1    ,  D      b    1    ,  D      c    1    )  ? ?b) If unit normal is   (  1      /    3  ,  2      /    3  ,  2      /    3  ) then       P    1   becomes   (  2      /    3  ,  4      /    3  ,  4      /    3  ) Where   D  =  2. We know that normal vector began on the plane at point       P    1   and ends at   (  1      /    3  ,  2      /    3  ,  2      /    3  )  . My questions is how unit normal vector coordinates value less than       P    1   coordinates value, because unit normal vector pointing outside of the plane it should be greater coordinates value than       P    1   ?

Mathias Patrick · Accepted Answer

Step 1The coordinates of a point P are the coordinates of the vector                     O        P            →       , with O the origin. The coordinates of a vector                     A        B            →       are the coordinates of its end point B minus the coordinates of its starting point A:                    A        B            →        =                    O        B            →        −                    O        A            →        .a) We are told that       P    1   is at distance D of the origin O and its direction is given by the vector                     n        →              1   . In other words,                     O                  P          1                    →        =  α                    n        →              1   for some positive real number   α . Since                     n        →              1   is a unit vector, we have  D  =  ‖                    O                  P          1                    →        ‖  =  α  ‖                    n        →              1    ‖  =  α  ,which implies                     O                  P          1                    →        =  D                    n        →              1   . Thus the coordinates of       P    1   are   (  D      a    1    ,  D      b    1    ,  D      c    1    ) with   (      a    1    ,      b    1    ,      c    1    ) the coordinates of                     n        1            →       .b) If                     n        →              1   starts at       P    1   and ends at a point       Q    1   , i.e.                    n        →              1    =                              P          1                          Q          1                    →       , then the coordinates of        Q    1   are the sum of the coordinates of       P    1   and                     n        →              1   , which gives   (  (  D  +  1  )      a    1    ,  (  D  +  1  )      b    1    ,  (  D  +  1  )      c    1    ) . If                     n        →              1   is   (  1      /    3  ,  2      /    3  ,  2      /    3  ) and   D  =  2 then       P    1   has coordinates   (  2      /    3  ,  4      /    3  ,  4      /    3  ) and                     n        →              1   ends at       Q    1   with coordinates   (  1  ,  2  ,  2  )

Plane with normal vector a) If unit normal vector is ( a 1 </msub> ,

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