קיץ 2021 · מועד ב

$= -\vec{v} + k\bigl(\vec{u} + \tfrac{1}{2}(\vec{AC} + \vec{CB})\bigr) = -\vec{v} + k\bigl(\vec{u} - \tfrac{1}{2}\vec{u} + \tfrac{1}{2}\vec{v}\bigr) = \tfrac{k}{2}\vec{u} + \bigl(\tfrac{k}{2} - 1\bigr)\vec{v}$

$\begin{cases} \tfrac{t}{3} = \tfrac{k}{2} \\ -t = \tfrac{k}{2} - 1 \end{cases} \;\Rightarrow\; \begin{cases} t = \tfrac{3k}{2} \\ t = 1 - \tfrac{k}{2} \end{cases}$

$\tfrac{3k}{2} = 1 - \tfrac{k}{2} \;\Rightarrow\; 2k = 1 \;\Rightarrow\; k = \tfrac{1}{2}$

$t = \tfrac{3}{2}\cdot\tfrac{1}{2} = \tfrac{3}{4}$

$E = (3,0,0) + (-2,\,4,\,0) = (1,\,4,\,0)$

$F = (0,0,12) + \tfrac{3}{4}\bigl((1,4,0) - (0,0,12)\bigr) = (0,0,12) + \tfrac{3}{4}(1,\,4,\,-12)$

$F = (0,0,12) + \bigl(\tfrac{3}{4},\,3,\,-9\bigr) = \bigl(\tfrac{3}{4},\,3,\,3\bigr)$

$\begin{cases} (a,b,c)\cdot(3,0,0) = 0 \\ (a,b,c)\cdot(1,4,0) = 0 \end{cases} \;\Rightarrow\; \begin{cases} 3a = 0 \\ a + 4b = 0 \end{cases} \;\Rightarrow\; a = 0\,,\; b = 0$

$0\cdot x + 0\cdot y + 1\cdot z = 0 \;\Rightarrow\; z = 0$

$|\vec{OA}| = 3$

$|\vec{OE}| = \sqrt{1^2 + 4^2} = \sqrt{17}$

$\cos(\angle AOE) = \dfrac{\vec{OA}\cdot\vec{OE}}{|\vec{OA}|\,|\vec{OE}|} = \dfrac{(3,0,0)\cdot(1,4,0)}{3\sqrt{17}} = \dfrac{3}{3\sqrt{17}} = \dfrac{1}{\sqrt{17}}$

$\sin(\angle AOE) = \sqrt{1 - \tfrac{1}{17}} = \dfrac{4}{\sqrt{17}} \quad (\angle AOE \approx 75.96°)$

$S_{AOE} = \tfrac{1}{2}\,|\vec{OA}|\,|\vec{OE}|\,\sin(\angle AOE) = \tfrac{1}{2}\cdot 3\cdot\sqrt{17}\cdot\dfrac{4}{\sqrt{17}} = 6$

$V_{FAOE} = \dfrac{S_{AOE}\cdot d}{3} = \dfrac{6\cdot 3}{3} = 6$