Given that [tex]\(\overrightarrow{PQ}\)[/tex] and [tex]\(\overrightarrow{RS}\)[/tex] intersect to form four right angles, let's analyze the provided statements to determine the correct one:
A. [tex]\(\overrightarrow{QQ}\)[/tex] and [tex]\(\overrightarrow{RS}\)[/tex] are parallel: This statement suggests that [tex]\(\overrightarrow{QQ}\)[/tex] and [tex]\(\overrightarrow{RS}\)[/tex] do not intersect and run in the same direction. However, [tex]\(\overrightarrow{QQ}\)[/tex] seems to indicate a zero vector (from point Q to itself), which is not parallel to any line. Therefore, this statement is invalid.
B. [tex]\(\overrightarrow{QQ}\)[/tex] and [tex]\(\overrightarrow{RS}\)[/tex] are skew: Skew lines are lines that do not intersect and are not parallel. Since [tex]\(\overrightarrow{QQ}\)[/tex] is a degenerate line (a point), it cannot be skew to any line. This statement is also invalid.
C. [tex]\(\overrightarrow{PQ} \perp \overrightarrow{RS}\)[/tex]: If [tex]\(\overrightarrow{PQ}\)[/tex] and [tex]\(\overrightarrow{RS}\)[/tex] intersect to form four right angles, then these lines must be perpendicular to each other. Perpendicularity ([tex]\(\perp\)[/tex]) indicates that the angle between the lines is 90 degrees, which matches the condition given in the problem.
D. [tex]\(\overrightarrow{PQ} = \overrightarrow{RS}\)[/tex]: This statement implies that the vectors [tex]\(\overrightarrow{PQ}\)[/tex] and [tex]\(\overrightarrow{RS}\)[/tex] are equal in magnitude and direction. However, since these lines intersect to form right angles, they cannot be the same line. Thus, this statement is incorrect.
Based on the above analysis, the correct statement is:
C. [tex]\(\overrightarrow{PQ} \perp \overrightarrow{RS}\)[/tex]
Therefore, the correct answer is C.
