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Math Help - Parallelepiped volumes

  1. #1
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    Parallelepiped volumes

    Hi there, pretty stuck on the following question from my exam revision:

    Find the volume of the following parallelepiped, one of whose vertices is the origin, and with the edges linking the origin to the points (co-ordinates in Metres) (1) (1) (7)
    (2) (0) (2)
    (2) (3) (-1)

    And also this, i think it is included:

    Find the point of intersection of the following line and plane:
    - The PLane passes through the points (3,3,4), (3,1,2) and (1,0,0)
    - The line passes through the points (3,5,8) and (7,13,22)

    THanks
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  2. #2
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by jameyt View Post
    Hi there, pretty stuck on the following question from my exam revision:

    Find the volume of the following parallelepiped, one of whose vertices is the origin, and with the edges linking the origin to the points (co-ordinates in Metres) (1) (1) (7)
    (2) (0) (2)
    (2) (3) (-1)

    And also this, i think it is included:

    Find the point of intersection of the following line and plane:
    - The PLane passes through the points (3,3,4), (3,1,2) and (1,0,0)
    - The line passes through the points (3,5,8) and (7,13,22)

    THanks
    This will start you off:
    Parallelepiped - Wikipedia, the free encyclopedia
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  3. #3
    MHF Contributor

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    Quote Originally Posted by jameyt View Post
    Hi there, pretty stuck on the following question from my exam revision:

    Find the volume of the following parallelepiped, one of whose vertices is the origin, and with the edges linking the origin to the points (co-ordinates in Metres) (1) (1) (7)
    (2) (0) (2)
    (2) (3) (-1)
    The "triple" product of three vectors (u \times v)\cdot w gives the volume of the parallelepiped having u, v, and w as edges. And that can be done most simply as a single determinant. If u= <u_x, u_y, u_z>, v= <v_x, v_y, v_z>, and w= <w_x, w_y, w_z> then (u\times v)\cdot w= \left|\begin{array}{ccc}u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z\end{array}\right|

    And also this, i think it is included:

    Find the point of intersection of the following line and plane:
    - The PLane passes through the points (3,3,4), (3,1,2) and (1,0,0)
    - The line passes through the points (3,5,8) and (7,13,22)

    THanks
    The vector from (3,3,4) to (3,1,2) is <0, -2, -2> and is a vector in the plane. The vector from (2,3,4) to (1, 0, 0) is <-1, -3, -4> and is another vector in the plane. Take the cross product of those two vectors to get a vector perpendicular to the plane. Can you write down the equation of a plane knowing one point and a perpendicular vector?

    The vector from (3, 5, 8) to (7, 13, 22) is <-5, -8, -14>. Can you write parametric equations for a line given a direction vector and one point?

    Put the parametric equations for x, y, and z into the equation for the plane and solve the equation for the parameter. The use that value to find the coordinates x, y, and z.
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