Answer
Please see the proof below.
Work Step by Step
When two vectors $u$ and $v$ are orthogonal, then the angle between these two vectors is given by $90^\circ$.
Thus, we have:
$||u\times v||=||u||\ ||v||\sin{\theta}\\=||u||\ ||v||\sin{90^\circ}\\=||u||\ ||v|| \ (1)\\=||u|| \ ||v||$
When two vectors $u$ and $v$ are unit vectors, then $||u|| \ ||v|| =1$
Therefore, $||u\times v||=||u||\ ||v||\\=(1)(1)\\=1$
This means that the cross product of two unit vectors is also a unit vector and has magnitude $1$.
