Math Help - Parameterisation

1. Parameterisation

Find a parameterisation of the circle of radius one formed by intersecting the plane $x+y+z=0$ and the sphere $x^2+y^2+z^2=1$.

(Hint: Start by finding two orthogonal unit vectors in the plane).

Starting with the hint, my first vector is $\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\
{-1}\\
0\\
\end{pmatrix}$
and my second vector is $\frac{1}{\sqrt{2}} \begin{pmatrix}
{-1}\\
1\\
0\\ \end{pmatrix}$

Where do I go from here?

2. Consider the unit circle: Center O (0,0), Right P (1,0), Up Q (0,1). The standard parameterization is $R(t)=O+\cos(t)P+\sin(t)Q=(\cos(t),\sin(t))$. The same formula works in your application: Find O,P,Q and use $R(t)=O+\cos(t)P+\sin(t)Q$