קיץ 2023 · מועד א

$\vec{CA'} = \vec{CB} + \vec{BA} + \vec{AA'} = -\vec v - \vec u + \vec w$

$\vec{BD} = \vec{BC} + \vec{CD} = \vec v - \vec u$

$\vec{BC'} = \vec{BC} + \vec{CC'} = \vec v + \vec w$

$\vec{CA'}\cdot\vec{BD} = (-\vec v - \vec u + \vec w)\cdot(\vec v - \vec u) = -|\vec v|^2 + |\vec u|^2 = 0$

$\vec{CA'}\cdot\vec{BC'} = (-\vec v - \vec u + \vec w)\cdot(\vec v + \vec w) = -|\vec v|^2 + |\vec w|^2 = 0$

$\vec{CE} = \vec{CC'} + \vec{C'E} = \vec w + \tfrac23\vec{C'F}$

$\vec{C'F} = \vec{C'C} + \vec{CB} + \tfrac12\vec{BD} = -\vec w - \vec v + \tfrac12\vec{BD}$

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

$\vec{C'F} = -\vec w - \vec v + \tfrac12(-\vec u + \vec v) = -\vec w - \tfrac12\vec u - \tfrac12\vec v$

$\vec{CE} = \vec w + \tfrac23\left(-\vec w - \tfrac12\vec u - \tfrac12\vec v\right) = \tfrac13\vec w - \tfrac13\vec v - \tfrac13\vec u$

$\vec{CE} = \tfrac13(\vec w - \vec v - \vec u) = \tfrac13\vec{CA'}$

$\vec{DA}\perp\vec{DC}$: $\;\vec{DA}\cdot\vec{DC} = 0$

$\vec{DA} = (3, n, p)$, $\;\vec{DC} = (4, 3, 0)$

$(3,n,p)\cdot(4,3,0) = 12 + 3n = 0 \Rightarrow n = -4$

$|\vec{DA}| = |\vec{DC}|$: $\;\sqrt{9 + 16 + p^2} = \sqrt{16 + 9}$

$25 + p^2 = 25$

$p^2 = 0 \Rightarrow p = 0$

$A(3, -4, 0)$

$|\vec{CC'}| = |\vec{DC}|$: $\;\sqrt{0 + 0 + z^2} = \sqrt{16 + 9}$

$z^2 = 25 \Rightarrow z = \pm 5$

$C' = C + (0,0,5) = (4, 3, 5)$

$B = (3,-4,0) + (4,3,0) = (7, -1, 0)$

$\ell:\;\; X = (7,-1,0) + t(-3,4,5)$

$X = (7,-1,0) + t(-3,4,5) + s(1,0,0)$