The matrix is as follows:
$$ \begin{bmatrix} 3/5 & 4/5 & 3/5 \\ -4/5 & 3/5 & 0 \\ 0 & 0 & 4/5 \\ \end{bmatrix} $$
I get that in order for a matrix (call this matrix A) to be orthogonal it first must be an nxn matrix and that $|AX|\ ∀X ∈ ℝ^n$ must be equal to $|X|$; along with $(A^t)(A) = I$ and the columns of $A$ form an orthonormal subset of $ℝ^n.$
keeping all of this in mind, what I first did was to just make sure and check to see if the magnitude of each column equaled $1$ and saw that they do indeed have a length of $1$, so that's good there. From here I took the dot product of each column and then saw that only $A_1$ dotted with $A_2$ works and have come to the conclusion that I need to change A_3, and this is where I'm stuck. Without doing what seems like endless trial and error, how does one solve this problem? I'm trying to keep all of these ideas in mind I'm just stuck in a rut it feels like. Thanks for the help!
Assume that last column is $$x=\left(\begin{array} ba \\ b \\ c \\ \end{array}\right).$$ Since third column is orthogonal to first two, we get $$3a/5-4b/5=0,$$ $$4a/5+3b/5=0$$ which implies that $a=0,b=0.$ Now magnitude of $x$ is $1$ gives that $c=1$. Hence we get $x$.