In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1. $\mathscr{P}$ denotes the plane $(AFH)$. $I$ is the midpoint of $[AE]$, $J$ is the midpoint of $[BC]$, $K$ is the midpoint of $[HF]$, $L$ is the intersection point of line $(EC)$ and plane $\mathscr{P}$. a. $\overrightarrow{AL} = \frac{1}{2}\overrightarrow{AH} + \frac{1}{2}\overrightarrow{AF}$. b. $\overrightarrow{AL} = \frac{1}{3}\overrightarrow{AK}$. c. $\overrightarrow{ID} = \frac{1}{2}\overrightarrow{IJ}$. d. $\overrightarrow{AL} = \frac{1}{3}\overrightarrow{AB} + \frac{1}{3}\overrightarrow{AD} + \frac{2}{3}\overrightarrow{AE}$.
In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1. $\mathscr{P}$ denotes the plane $(AFH)$. $I$ is the midpoint of $[AE]$, $J$ is the midpoint of $[BC]$, $K$ is the midpoint of $[HF]$, $L$ is the intersection point of line $(EC)$ and plane $\mathscr{P}$.
a. $\overrightarrow{AL} = \frac{1}{2}\overrightarrow{AH} + \frac{1}{2}\overrightarrow{AF}$.\\
b. $\overrightarrow{AL} = \frac{1}{3}\overrightarrow{AK}$.\\
c. $\overrightarrow{ID} = \frac{1}{2}\overrightarrow{IJ}$.\\
d. $\overrightarrow{AL} = \frac{1}{3}\overrightarrow{AB} + \frac{1}{3}\overrightarrow{AD} + \frac{2}{3}\overrightarrow{AE}$.