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Math Help - equation

  1. #1
    Member Veronica1999's Avatar
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    equation

    A bird flies linearly, according to the equation
    (x,y,z) = (5,6,7) + t[2,3,1]. Assume that the Sun is directly overhead , making the Sun's rays perpendicular to the xy-plane which represents the ground. The birds shadow is said to be projected perpendicularly onto the level ground. Find an equation that describes the motion of the shadow.


    I got y=3/2x- 3/2

    I am not sure about my answer.
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  2. #2
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    Hello, Veronica1999!

    A bird flies linearly, according to the equation: . (x,y,z) \:=\: (5,6,7) + t[2,3,1]
    Assume that the Sun is directly overhead, making the Sun's rays perpendicular
    to the xy-plane, which represents the ground.
    The bird's shadow is said to be projected perpendicularly onto the level ground.
    Find an equation that describes the motion of the shadow.

    I got: . y\:=\:\tfrac{3}{2}x - \tfrac{3}{2} . Correct!

    We have: . \begin{Bmatrix}x &=& 5 + 2t & [1] \\ y &=& 6+3t & [2] \\ z &=& 0 & [3] \end{Bmatrix}

    From [1]: . x \:=\:5+2t \quad\Rightarrow\quad t \:=\:\frac{x-5}{2}\;\;[4]

    From [2]: . y \:=\:6+3t \quad\Rightarrow\quad t \:=\:\frac{y-6}{3}\;\;[5]


    Equate [4] and [5]: . \frac{x-5}{2} \:=\:\frac{y-6}{3} \quad\Rightarrow\quad y \:=\:\tfrac{3}{2}x - \tfrac{3}{2}

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  3. #3
    MHF Contributor

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    I did this in a slightly different way (only slightly):
    From x= 5+ 2t, t= (x- 5)/2.

    Then y= 6+ 3t= 6+ 3(x- 5)/2= 6+ (3/2)x- 15/2= 12/2+ (3/2)x- 15/2= (3/2)x- 3/2

    Saying that the projection is perpendicular means that we can just ignore z.
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