קיץ 2023 · מועד מיוחד

$\vec{BD'} = \vec{BA} + \vec{AD} + \vec{DD'} = -\vec u + \vec v + \vec w$

$\vec{FE} = \vec{FD'} + \vec{D'E} = (1-t)(-\vec u+\vec v+\vec w) - \frac25\vec w$

$\vec{FE} = (t-1)\vec u + (1-t)\vec v + \left(\tfrac35-t\right)\vec w$

$\tfrac35 - t = 0 \Rightarrow t = \tfrac35$

$t=\frac35$: $\;\vec{BF} = \frac35\vec{BD'} = \frac35(-\vec u+\vec v+\vec w)$

$C(0,0,0)$, $B(x,0,0)$, $D(0,y,0)$, $D'(0,y,z)$; $\;\vec{BD'} = (-x,y,z)$

$\vec{BF} = \frac35(-x,y,z) = \left(-\tfrac35 x, \tfrac35 y, \tfrac35 z\right)$

$\vec{BF} = F-B = (4-x,\,12,\,18)$

$4-x = -\tfrac35 x \Rightarrow x=10$

$12 = \tfrac35 y \Rightarrow y=20$

$18 = \tfrac35 z \Rightarrow z=30$

$B(10,0,0)$, $D(0,20,0)$, $D'(0,20,30)$

$|\vec u| = |AB| = |DC| = 20$

$|\vec v| = |AD| = |BC| = 10$

$|\vec w| = |DD'| = 30$

$\vec{BD} = D-B = (-10,20,0)$, $\;\vec{BF} = F-B = (-6,12,18)$

$(a,b,c)\cdot(-6,12,18) = -6a+12b+18c = 0$

$18c = 0 \Rightarrow c = 0$

$P = (4,12,18) - 2(2,1,0) = (0,\,10,\,18)$