The terminal side of an angle $\alpha$ in standard position intersects the unit circle at $\left(-\frac{\sqrt{2}}{3}, \frac{\sqrt{7}}{3}\right)$. Find $\cos \alpha$

Step 1 ：The terminal side of an angle \(\alpha\) in standard position intersects the unit circle at \(\left(-\frac{\sqrt{2}}{3}, \frac{\sqrt{7}}{3}\right)\).

Step 2 ：The x-coordinate of the point where the terminal side of the angle intersects the unit circle is equal to \(\cos \alpha\).

Step 3 ：So, \(\cos \alpha = -\frac{\sqrt{2}}{3}\).

Step 4 ：\(\boxed{-\frac{\sqrt{2}}{3}}\) is the final answer.