Answer
$$[x]_B=\left[ 11 \quad 2 \quad 15\right]^T.$$
Work Step by Step
Let $x=(5,10,15)$ and $B=\{ (\frac{3}{5},\frac{4}{5},0),(-\frac{4}{5},\frac{3}{5},0),(0,0,1) \}$
To find the coordinates of $x$ relative to $B$, we have to find the following;
$$\langle (5,10,15),(\frac{3}{5},\frac{4}{5},0)\rangle = 3+8=11$$
$$\langle (5,10,15),(-\frac{4}{5},\frac{3}{5},0)\rangle = -4+6=2$$
$$\langle (5,10,15),(0,0,1)\rangle = 15.$$
Then, the coordinate matrix of $x$ relative to $B$ is given by
$$[x]_B=\left[ 11 \quad 2 \quad 15\right]^T.$$
