Random Vectors

Discrete Mathematics Level 3

Consider two unit vectors $$\vec{u}_1, \vec{u}_2$$ chosen uniformly at random from all unit vectors in $$\mathbb{R}^3$$ such that $$\vec{u}_1 \cdot \hat{k} = \vec{u}_2 \cdot \hat{k} = 0$$. What is the expected value of $$|u_1 + u_2|$$?

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