If the planes x = c y + b z , y = c x + a z , z = b x + a y go through the straight line, then is it true that a 2 + b 2 + c 2 + 2 a b c = 1 ?

Question

If the planes       x    =    c    y    +    b    z   ,       y    =    c    x    +    a    z   ,       z    =    b    x    +    a    y   go through the straight line, then is it true that             a      2        +          b      2        +          c      2        +    2    a    b    c    =    1  ?

Kira Ramos · Accepted Answer

Let the planes       x    =    c    y    +    b    z   ,       y    =    c    x    +    a    z   ,       z    =    b    x    +    a    y   go through the straight line defined by             (      p      ,      q      ,      r      )      . The planes can also be written as      x    -    c    y    -    b    z    =    0   ,       c    x    -    y    +    a    z    =    0   ,       b    x    +    a    y    -    z    =    0  As the aircraft fly through the line             (      p      ,      q      ,      r      )      , the line is perpendicular to the normal of the plane, say       x    -    c    y    -    b    z    =    0   and hence dot product should be zero i.e.      p    -    c    q    -    b    r    =    0  Similarly       c    p    -    q    +    a    r    =    0   and      b    p    +    a    q    -    r    =    0  Solving them for       p    ,    q   and       r   from first two equations, we get            p              -        a        c        -        b              =                  -        q                    a        +        b        c              =          r              -        1        +                  c          2                    which implies       p    =    -    a    c    -    b  ,       q    =    -    a    -    b    c   and       r    =          c      2        -    1  and by adding third we obtain      b          (      -      a      c      -      b      )        +    a          (      -      a      -      b      c      )        -          (              c        2            -      1      )        =    0  or       -    a    b    c    -          b      2        -          a      2        -    a    b    c    -          c      2        +    1    =    0  or             a      2        +          b      2        +          c      2        +    2    a    b    c    =    1

If the planes x=cy+bz , y=cx+az , z=bx+ay go through the straight line, then is it true that a^2+b^2+c^2+2abc=1?

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