taiwan-gsat 2023 Q6

taiwan-gsat · Other · gsat__math-a 5 marks Vectors Introduction & 2D Dot Product Computation

In coordinate space, consider a unit cube with edge length 1, with one vertex $O$ fixed. From the seven vertices other than $O$, two distinct points are randomly selected, denoted as $P$ and $Q$. What is the expected value of the dot product $\overrightarrow{OP} \cdot \overrightarrow{OQ}$ among the following options?
(1) $\frac{4}{7}$
(2) $\frac{5}{7}$
(3) $\frac{6}{7}$
(4) 1
(5) $\frac{8}{7}$

In coordinate space, consider a unit cube with edge length 1, with one vertex $O$ fixed. From the seven vertices other than $O$, two distinct points are randomly selected, denoted as $P$ and $Q$. What is the expected value of the dot product $\overrightarrow{OP} \cdot \overrightarrow{OQ}$ among the following options?\\
(1) $\frac{4}{7}$\\
(2) $\frac{5}{7}$\\
(3) $\frac{6}{7}$\\
(4) 1\\
(5) $\frac{8}{7}$

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