The spherical distance between two points $P_1=(0,0,1)$ and $P_2=(\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}},{-}\frac{\sqrt{3}}{2})$ is $\frac{5\pi }{6}$.
I am at a loss as to how the spherical distance was obtained. My notes just give the answer but no step by step derivation. Any help would be much appreciated.
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Given two points on a sphere, the shortest line connecting them is an arc of a great circle, a 'great circle' being the intersection of the circle with a plane that goes through the circle's center. The distance between two points on a sphere is the length of that arc.
Since your points have length 1, they are on the unit sphere $\{(x, y, z) \mid x^2 + y^2 + z^2 = 1\}$, and any arc connecting them has length equal to the angle between them (in general, remember, the length of an arc is $r\theta$, the product of radius and subtended angle).
Combining this with Martín's answer, you know $P_1 \cdot P_2 = \frac{-\sqrt{3}}{2}$, and the angle between them is $\cos^{-1}\left(\frac{-\sqrt3}{2}\right)$.
Idea: the points $P_1, P_2$ are also unit vectors, so $P_1\cdot P_2= \cos(\text{angle between $P_1$ and $P_2$})$.