Question 57
If $u_1, \dots, u_k \in \mathbb{R}^3$, denote by $C(u_1, \dots, u_k)$ the cone generated by $u_1, \dots, u_k$:
$$
C(u_1, \dots, u_k) = \{a_1 u_1 + \dots + a_k u_k; a_1, \dots, a_k \in [0, +\infty)\}.
$$
Let $v_1, v_2, v_3, v_4$ points randomly and independently chosen from the unit sphere $x^2 + y^2 + z^2 = 1$.
a. What is the probability that $C(v_1, v_2, v_3, v_4) = \mathbb{R}^3$?
b. What is the probability that each of the vectors is needed to generate $C(v_1, v_2, v_3, v_4)$, i.e., that $C(v_1, v_2, v_3) \neq C(v_1, v_2, v_3, v_4)$, $C(v_1, v_2, v_4) \neq C(v_1, v_2, v_3, v_4)$, $C(v_1, v_3, v_4) \neq C(v_1, v_2, v_3, v_4)$ and $C(v_2, v_3, v_4) \neq C(v_1, v_2, v_3, v_4)$?
$$
C(u_1, \dots, u_k) = \{a_1 u_1 + \dots + a_k u_k; a_1, \dots, a_k \in [0, +\infty)\}.
$$
Let $v_1, v_2, v_3, v_4$ points randomly and independently chosen from the unit sphere $x^2 + y^2 + z^2 = 1$.
a. What is the probability that $C(v_1, v_2, v_3, v_4) = \mathbb{R}^3$?
b. What is the probability that each of the vectors is needed to generate $C(v_1, v_2, v_3, v_4)$, i.e., that $C(v_1, v_2, v_3) \neq C(v_1, v_2, v_3, v_4)$, $C(v_1, v_2, v_4) \neq C(v_1, v_2, v_3, v_4)$, $C(v_1, v_3, v_4) \neq C(v_1, v_2, v_3, v_4)$ and $C(v_2, v_3, v_4) \neq C(v_1, v_2, v_3, v_4)$?