The angle between two vectors will be deferred by a single point, which is called as the shortest angle at which we have to turn around one of the vectors to the position of co-directional with another vector.

The extreme vectors of Sofia bats are $\vec u = < -8, 12 >$ and $\vec v = < -8, 12 >$

The angle between the vectors is

$\theta = cos^{-1} \frac{\vec u.\vec v}{||\vec u|| ||\vec v||}$

The dot product is calculated as

$\vec u. \vec v = (-8)(13)+(12)(15)$

⇒ $-104+180$

⇒ 76

The magnitude can be calculated as

$||\vec u|| = \sqrt{(-8 )^{2} +(12)^{2} }$

⇒ $\sqrt{64+144}$

⇒ $\sqrt{208}$

$||\vec v|| = \sqrt{(13 )^{2} +(15)^{2} }$

⇒ $\sqrt{169+225}$

⇒ $\sqrt{394}$

Thus the angle between the vectors is

$\theta = cos^{-1} \frac{76}{(\sqrt{208} )(\sqrt{394} )}$

⇒ $\theta = cos^{-1} \frac{76}{\sqrt{81952} }$

⇒ $\theta = cos^{-1} \frac{76}{286.27 }$

⇒ $\theta = cos^{-1} (0.2654)$

⇒ $\theta=74.60$

Hence we can conclude that the angle between the two extreme vectors of Sofia bats is 74.60 degrees.

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