Question 1163487
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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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<pre>
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(5^2+3^2)}}} = {{{sqrt(25+9)}}} = {{{sqrt(34)}}}.



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

    {{{tan(alpha)}}} = {{{2/sqrt(34)}}}.


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

    {{{tan(alpha)}}} = {{{8/L}}},



which implies for the shadow length

     L = {{{8/(tan(alpha))}}} = {{{8/((2/sqrt(34)))}}} = {{{4*sqrt(34)}}} = 23.32 meters (rounded).    <U>ANSWER</U>
</pre>

Solved.