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]?
This problem confuses me. Doesn't the location of the pole matter?
let the pole be at the origin of the xy plane, represented by the vector $\displaystyle <0,0,8>$
let the vector $\displaystyle <5d,3d,-2d>$ represent a sun ray vector from the top of the pole to the ground.
$\displaystyle <0,0,8> + <5d,3d,-2d> = <5d,3d,8-2d>$ ... the resulting shadow vector lying on the xy plane.
since the shadow vector lies on the ground, $\displaystyle 8-2d = 0$ ...
<0,0,8> + <20,12,-8> = <20,12,0>
magnitude of the shadow vector ...
$\displaystyle \sqrt{20^2+12^2+0^2} = 4\sqrt{34} \approx 23.3 \, m$
Well, there is one piece of information which is not explicitly given in that assignment: the Sun's rays are all parallel with each other (because the Sun is very far away from us). That's why it doesn't matter where on the xy-plane the pole is.