Convert the point from rectangular coordinates into polar coordinates with $r \geq 0$ and $0 \leq \theta<2 \pi$. \[ (7,-7) \]

Step 1 ：Given the rectangular coordinates of a point as (7,-7).

Step 2 ：We need to convert these rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\).

Step 3 ：The polar coordinates of a point in the plane are given by \((r, \theta)\) where \(r\) is the distance of the point from the origin and \(\theta\) is the angle the line joining the point to the origin makes with the positive x-axis.

Step 4 ：The conversion from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\) is given by: \[r = \sqrt{x^2 + y^2}\] and \[\theta = \arctan\left(\frac{y}{x}\right)\].

Step 5 ：However, the \(\arctan\) function only gives values in the range \((-\pi/2, \pi/2)\), so we need to adjust the value of \(\theta\) depending on the quadrant of the point.

Step 6 ：If \(x > 0\) and \(y \geq 0\), \(\theta\) is in the correct range.

Step 7 ：If \(x > 0\) and \(y < 0\), we need to add \(2\pi\) to \(\theta\) to bring it into the range \((0, 2\pi)\).

Step 8 ：If \(x < 0\), we need to add \(\pi\) to \(\theta\) to bring it into the correct range.

Step 9 ：If \(x = 0\) and \(y > 0\), \(\theta = \pi/2\).

Step 10 ：If \(x = 0\) and \(y < 0\), \(\theta = 3\pi/2\).

Step 11 ：For the given point \((7,-7)\), we have \(x = 7\) and \(y = -7\).

Step 12 ：Calculating \(r\) using the formula, we get \(r = \sqrt{7^2 + (-7)^2} = 9.899494936611665\).

Step 13 ：Calculating \(\theta\) using the formula, we get \(\theta = \arctan\left(\frac{-7}{7}\right)\). Since \(x > 0\) and \(y < 0\), we need to add \(2\pi\) to \(\theta\) to bring it into the range \((0, 2\pi)\). So, \(\theta = 5.497787143782138\).

Step 14 ：\(\boxed{\text{Final Answer: The polar coordinates of the point }(7,-7)\text{ are }\left(9.899494936611665, 5.497787143782138\right)}\)