קיץ 2021 · מועד מיוחד

$\vec{AA'} \cdot \vec{AB} = 0$

$\vec{AA'} \cdot \vec{AC} = 0$

$(k-1)(k-1) + (k-7)(k) + (k+1)(3) = 0$

$k^2 - 2k + 1 + k^2 - 7k + 3k + 3 = 0$

$2k^2 - 6k + 4 = 0 \;\Longrightarrow\; k^2 - 3k + 2 = 0$

$(k-1)(k+1) + (k-7)\cdot 0 + (k+1)(k-3) = 0$

$k^2 - 1 + k^2 - 2k - 3 = 0$

$2k^2 - 2k - 4 = 0 \;\Longrightarrow\; k^2 - k - 2 = 0$

$k = 2$

$\vec{AA'} = (1,-5,3),\quad \vec{AB} = (1,2,3),\quad \vec{AC} = (3,0,-1)$

$\ell_{AC}:\; X = (8,-1,-1) + t(3,0,-1)$

$\ell_{BC}:\; X = (4,0,2) + m(2,-2,-4)$

$(8,-1,-1) + t(3,0,-1) = (4,0,2) + m(2,-2,-4)$

$\begin{cases} 8+3t = 4+2m \\ -1 = -2m \\ -1-t = 2-4m \end{cases}$

$C = (8,-1,-1) + (-1)(3,0,-1) = (5,-1,0)$

$x - 5y + 3z + d = 0$

$6 - 5\cdot(-6) + 3\cdot 3 + d = 0$

$6 + 30 + 9 + d = 0 \;\Longrightarrow\; d = -45$

$x - 5y + 3z - 45 = 0$

$\angle C'A'B' = \angle CAB$

$\cos(\angle CAB) = \dfrac{\vec{AC}\cdot\vec{AB}}{|\vec{AC}|\cdot|\vec{AB}|}$

$\vec{AC}\cdot\vec{AB} = (3,0,-1)\cdot(1,2,3) = 3 + 0 - 3 = 0$

$\angle C'A'B' = 90°$

$A = C - \vec{AC} = (5,-1,0) - (3,0,-1) = (2,-1,1)$

$B = A + \vec{AB} = (2,-1,1) + (1,2,3) = (3,1,4)$

$B' = B + \vec{AA'} = (3,1,4) + (1,-5,3) = (4,-4,7)$

$x_M = \dfrac{4+6}{2} = 5$

$y_M = \dfrac{-4+(-6)}{2} = -5$

$z_M = \dfrac{7+3}{2} = 5$

$M = (5,-5,5)$