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Math Help - Point symmetrical

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    Point symmetrical

    Finding the coordinates of the point Q, symmetrical from the point P (2,1,3) in relation to the plane 4x -3y + z +18 = 0



    Answer:
    (-6,7,1)
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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    Finding the coordinates of the point Q, symmetrical from the point P (2,1,3) in relation to the plane 4x -3y + z +18 = 0



    Answer:
    (-6,7,1)
    1. Calculate the equation of a line l through P perpendicular to the given plane E: (4,-3,1) \cdot (x,y,z)+18=0

    l: (x,y,z)=(2,1,3)+t \cdot (4,-3,1)

    2. Let M denote the intersection point between l and the given plane:

    E \cap l = \{M\}~\implies~ (4,-3,1) \cdot \left((2,1,3)+t \cdot (4,-3,1)\right)+18=0 ~\implies~8+26t+18=0~\implies~ t = -1

    and therefore M(-2,4,2)

    3. M is the midpoint of \overline{PQ} . Let Q(q_1 ,q_2 ,q_3 ) denote the reflection point of P with respect to E. Then you know by the formula calculating the midpoint between 2 points:

    \frac{2+q_1}{2} =-2 \ \wedge\ \frac{1+q_2}{2} =4 \ \wedge\ \frac{3+q_3}{2} =2

    Solve these equations for q_i. You'll get Q(-6,7,1)
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    Thank you
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