If $\{u,v\}$ is an orthonormal set in an inner product space, then find $\|6u-8v\|$.
That's pretty much it. I'm trying to study for a quiz and can't figure it out. There are no examples in my book to help me out, just a question. I know that if $\{u,v\}$ are orthonormal then $\|u\|=1$ and $\|v\|=1$, but how does that translate into $\|6u-8v\|$? Thanks in advance to anyone that can help!
Compute the square of the norm and use the definition of the inner product as being bilinear:
$\|6u-8v\|^2 = \langle 6u-8v, 6u-8v \rangle = 36\langle u,u \rangle - 48\langle u, v \rangle - 48 \langle v,u \rangle + 64 \langle v,v \rangle$.
Now use that $u$ and $v$ are orthonormal and the fact that $\langle w,w \rangle$ is the squared length of the vector $w$. Take the square root to obtain the answer.