Final answer:
Rotating a line 90 degrees clockwise around the origin alters the positions of each point on the plane according to the rule (x, y) to (y, -x), while preserving the line's length and shape due to invariance under rotation.
Step-by-step explanation:
When a line is rotated 90 degrees clockwise about the origin, each point on the line will move to a new position on the plane. For example, a point with coordinates (x, y) would be transformed to the coordinates (y, -x). This is because rotating a point 90 degrees clockwise is equivalent to reflecting the point over the x-axis and then over the y-axis.
Since rotations preserve distances between points, the length of any segment of the line remains unchanged. The concept of invariance under rotation assures that the magnitudes of vectors remain constant despite the rotation. Hence, the overall shape and size of the line are maintained, only its orientation and position change. In coordinate system notations, if we have a system S and rotate it to become S', the coordinates change as a result of the rotation.
An increase in the slope of a line corresponds to a counter-clockwise rotation around the y-intercept, whereas increasing the y-intercept moves the line horizontally right without affecting its slope. Torque in physics also relates to directional rotation, being clockwise or counterclockwise depending on the pivot. However, in the context of a line rotation on a plane in mathematics, torque is not directly relevant.