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Math Help - Perpendicular vectors

  1. #1
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    Perpendicular vectors

    1. Given the points O (0, 0, 0), P (2, 6, −1), Q (1, 1, 1) and R (4, 6, 2), find a vector that is perpendicular to both

    (a) OP and OQ (b) PQ and QR (c) RP and QR

    2. Find an equation of the plane passing through point P and having vector n as a normal.
    P (−1, 3, −2); n = (-2,1,-1)

    3. Find an equation of the plane that passes through the given points.
    P (5, 4, 3), Q (4, 3, 1), R (1, 5, 4)






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  2. #2
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    Quote Originally Posted by dorwei92 View Post
    1. Given the points O (0, 0, 0), P (2, 6, −1), Q (1, 1, 1) and R (4, 6, 2), find a vector that is perpendicular to both

    (a) OP and OQ (b) PQ and QR (c) RP and QR

    2. Find an equation of the plane passing through point P and having vector n as a normal.
    P (−1, 3, −2); n = (-2,1,-1)

    3. Find an equation of the plane that passes through the given points.
    P (5, 4, 3), Q (4, 3, 1), R (1, 5, 4)






    1. Use the cross product:

    \vec n = \overrightarrow{OP} \times \overrightarrow{OQ}

    \vec n = (\overrightarrow{OQ}-\overrightarrow{OP}) \times (\overrightarrow{OR} - \overrightarrow{OQ})

    and so on

    2. If \vec x = (x,y,z) and \vec p is the staionary vector of P then the plane in question has the equation:

    \vec n(\vec x - \vec p)=0

    Plug in the given values.

    3. The plane has the equation:

    \vec x = \vec p + s \cdot (\vec q- \vec p) + t \cdot (\vec r - \vec p)
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  3. #3
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    I'm thankful for the reply but i still dont understand part 2 and 3 solution, could u explain it in a simpler manner?
    as in what does vector p is stationary vector of P and
    why is there a vector n beside it?

    for part 3;
    i totally got no idea what it means
    with the equation..i know nothing about it.
    so sorry, could u explain more specifically?
    thanks and sorry for the troubles.
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  4. #4
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    Quote Originally Posted by earboth View Post
    ...
    2. If \vec x = (x,y,z) and \vec p is the staionary vector of P then the plane in question has the equation:

    \vec n(\vec x - \vec p)=0

    Plug in the given values.

    Quote Originally Posted by dorwei92 View Post
    I'm thankful for the reply but i still dont understand part 2 and 3 solution, could u explain it in a simpler manner?
    as in what does vector p is stationary vector of P and
    why is there a vector n beside it?
    \vec n is the given normal vector of the plane; (\vec x - \vec p) describes any vector which is placed in the plane and therefore the dotproduct of these 2 vectors must be zero. Thus you get:

    (-2,1,-1) \cdot ((x,y,z)-(-1,2,-3))=0 Multiply the brackets:

    -2(x+1)+1(y-2)-1(z-3)=0~\implies~-2x+y-z=1

    for part 3;
    i totally got no idea what it means
    with the equation..i know nothing about it. <<<<<<<< what exactly have you done in vector geometry so far?
    so sorry, could u explain more specifically?
    thanks and sorry for the troubles.
    3. The plane has the equation:

    \vec x = \vec p + s \cdot (\vec q- \vec p) + t \cdot (\vec r - \vec p)
    Two points define a vector (and of course a straight line), three points define a plane. To determine a plane you need one fixed point P and two different directions (exactly: non-collinear). Then any point of the plane (which has the coordinates (x, y, z)) is described by:

    (x,y,z)= \underbrace{(5,4,3)}_{\vec p} + s \cdot (\underbrace{(4,3,1)}_{\vec q}-\underbrace{(5,4,3)}_{\vec p}) + t \cdot (\underbrace{(1,5,4)}_{\vec r}-\underbrace{(5,4,3)}_{\vec p})

    (x,y,z)= (5,4,3) + s \cdot (-1,-1,-2) + t \cdot (-4,1,1)
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  5. #5
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    thanks..now i really do understand quite a fair bit.
    but can u explain what does the s and t means
    for the equation of the plane in part 3?
    thanks^^
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  6. #6
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    Quote Originally Posted by dorwei92 View Post
    thanks..now i really do understand quite a fair bit.
    but can u explain what does the s and t means
    for the equation of the plane in part 3?
    thanks^^
    s, t are real numbers which you can choose to calculate the coordinates (or the staionary vector) of a point located in the plane.

    For instance s = 2 and t = -1 produce the point P(7, 1, -2)
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