Recall that ||x||² = x • x for any vector x, so
||u + v||² = (u + v) • (u + v)
||u + v||² = (u • u) + (u • v) + (v • u) + (v • v)
||u + v||² = ||u||² + 0 + 0 + ||v||²
||u + v||² = ||u||² + ||v||²
where u • v = v • u = 0 since u and v are perpendicular.
Then
||u + v||² = (5√2)² + (6√2)²
||u + v||² = 25•2 + 36•2
||u + v||² = 122
||u + v|| = √122
