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Math Help - Vectors - Parametric Equations

  1. #1
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    Question Vectors - Parametric Equations

    Earth is described by the equation : x + y + z = 18
    Mars is described by the equation: 4x + 3y - z = -3

    Earth and Mars intersect each other

    a/ find a parametric equation of the line in which earth intersect mars

    b/ The coordinate of Spring Lake is given by the point (-5, 10, 13)
    How far away from spring lake from the line mentioned in (a)?
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  2. #2
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    Hello, shiiganB!

    Astronomically, the problem makes no sense,
    . . but I'll solve it anyway.


    \begin{array}{ccccc}\text{Earth is described by the equation:} & x + y + z &=& 18 & [1] \\<br />
\text{Mars is described by the equation:} & 4x + 3y - z &=& \text{-}3& [2] \end{array}

    Earth and Mars intersect each other.

    (a) Find a parametric equation of the line in which Earth intersect Mars.

    Add [1] and [2]: . 5x + 4y \:=\:15 \quad\Rightarrow\quad x \:=\:3-\tfrac{4}{5}y

    Substitute into [1]: . \left(3-\tfrac{4}{5}y\right) + y + z \:=\:18 \quad\Rightarrow\quad z \:=\:15-\tfrac{1}{5}y

    . . We have: . \begin{Bmatrix}x &=& 3-\frac{4}{5}y \\ y &=& y \\ z &=& 15-\frac{1}{5}y \end{Bmatrix}


    On the right, replace y with the parameter t.

    . . Therefore: . \begin{Bmatrix}x &=& 3 - \frac{4}{5}t \\ y &=& t \\ z &=& 15 - \frac{1}{5}t \end{Bmatrix}




    (b) The coordinates of Spring Lake is given by the point (-5, 10, 13).
    How far away from Spring Lake is the line mentioned in (a)?
    A strange (silly!) question . . .

    Spring Lake could be a ski resort located on Mars or Jupiter or . . .

    Ignoring that possibility, Spring Lake is probably located on Earth,
    . . in which case, it is on the line mentioned in (a).

    . . The distance is zero.


    To check, try t = 10.

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