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Math Help - Vectors - Lines and Planes Question

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    Question Vectors - Lines and Planes Question

    According to an astronomy club, Earth is a plane described by the equation
    x + y + z = 18.

    The club also stated that Earth will be destroyed by an explosion that will spontaneously occur at point A = (1, 1, 1). Note that point A is not a point on Earth, it is a point in space that will explode and will impact Earth.

    a/ Calculate the co-ordinate of the first point that is going to be destroyed by this explosion.

    b/ What is the distance between point A and the first point that is going to be destroyed (mentioned in a)? (Hint: Shortest Distance)
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    Quote Originally Posted by shiiganB View Post
    According to an astronomy club, Earth is a plane described by the equation
    x + y + z = 18.

    The club also stated that Earth will be destroyed by an explosion that will spontaneously occur at point A = (1, 1, 1). Note that point A is not a point on Earth, it is a point in space that will explode and will impact Earth.

    a/ Calculate the co-ordinate of the first point that is going to be destroyed by this explosion.

    b/ What is the distance between point A and the first point that is going to be destroyed (mentioned in a)? (Hint: Shortest Distance)
    We want to find the line perpendicular to the plane passing through the point; this will tell us the minimal distance between the point and the plane.

    For a plane ax + by + cx + d = 0, we have normal vector \vec n = (a, b, c).

    The vector equation of the line passing through point P_0 with direction vector \vec v is

    \vec r(t)=P_0+t\cdot \vec v

    In this case, write

    \vec n = (1, 1, 1)

    \vec r(t)=(1,1,1)+t\cdot(1,1,1)

    This can also be written

    \vec r(t)=(1+t,1+t,1+t)

    In other words, we have a parametric equation described by

    (x,y,z)=(1+t,1+t,1+t)

    Combine with x+y+z=18 to get

    (1+t)+(1+t)+(1+t)=18

    t=5

    So the point that gets hit first is (6,6,6), and to find the distance, use the Pythagorean theorem (generalized for three dimensions):

    D^2 = 5^2 + 5^2 + 5^2

    D = 5\sqrt{3}

    Note that the line we found earlier also passes through the origin, so it could be expressed more concisely as

    \vec r(t)=(t,t,t)
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