SOLUTION: How long is the shadow cast on the ground (represented by the xy-plane) by a pole that is eight meters tall, given that the sun’s rays are parallel to the vector [5, 3, −2]?

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Question 1163487: How long is the shadow cast on the ground (represented by the xy-plane) by a pole that is eight meters tall, given that the sun’s rays are parallel to the vector [5, 3, −2]?
Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
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How long is the shadow cast on the ground (represented by the xy-plane) by a pole that is eight meters tall,
given that the sun’s rays are parallel to the vector [5, 3, −2]?
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First, the projection of the given 3D-vector to the (x,y)-plane is the vector V = [5,3].

It defines the direction to the sun in this plane, and its length is  | V | = sqrt%285%5E2%2B3%5E2%29 = sqrt%2825%2B9%29 = sqrt%2834%29.



Next, if the angle of sun rays with the Earth surface (which we interpret as the plane (x,y)) is alpha,  then

    tan%28alpha%29 = 2%2Fsqrt%2834%29.


From the other side, if L is the shadow length, then

    tan%28alpha%29 = 8%2FL,



which implies for the shadow length

     L = 8%2F%28tan%28alpha%29%29 = 8%2F%28%282%2Fsqrt%2834%29%29%29 = 4%2Asqrt%2834%29 = 23.32 meters (rounded).    ANSWER

Solved.