Q:

Let u and v be two orthonormal vectors in $\mathbb{R}^n$. Then $\| u+v \| $ is

Select one:
a. 2
b. √n
c. √2
d. n
I'm getting b but i'm confused 

A:

Let $u,v \in \mathbb{R}^n$. First recall that 
$$ ||u||^2 = u \cdot u$$
$u$ and $v$ are orthonormal if:
\begin{align}
||u|| &= 1\\
||v|| &= 1\\
u \cdot v &= 0
\end{align}
Then,
\begin{align}
||u + v||^2 &= (u + v) \cdot (u + v)\\
&= u \cdot u + 2 (u \cdot v) + v \cdot v\\
&= 1 + 2(0) + 1\\
&= 2\\
\Rightarrow ||u + v|| &= \sqrt{2}
\end{align}

