How to verify this is an orthogonal basis? How to transform it into an orthonormal basis? 
Let
  $$B = \left\{ \begin{bmatrix} 3\\ -3\\ 0\end{bmatrix},\begin{bmatrix} 2\\ 2\\ -1\end{bmatrix},\begin{bmatrix} 1\\ 1\\ 4\end{bmatrix}\right\},\qquad v =\begin{bmatrix} 5\\ -3\\ 1\end{bmatrix}.$$
  a) Verify that $B$ is an orthogonal basis of $\mathbb{R}^3$.
  b) Transform $B$ into an orthonormal basis.
  c) Write $v$ as linear combination of $B$.

I am really lost in class. I don't even know where to start. Please show steps and answers for the exercise problem so that I can learn. Thank you

Hints: 
a) Take the dot product of the vectors and it should be zero for any two different vectors. 
b) Divide each vector by its length.
c) $v = \alpha b_1 + \beta b_2 + \gamma b_3 $
To determine the $\alpha,\beta$, and $\gamma$, take the dot product of $v$ with $b_1,b_2$, and $b_3$ and note that $b_i.||b_i||=1,\, \forall i=1..3$. Note that, $\alpha,\beta$, and $\gamma$ are known as the Fourier coefficients.    
