Spin of a line

Spin line r, through P(5, 7, 10) with direction (2, 3, 2), by 30° around the axis through O(5, 7, 10) with direction (1, 0, 0).

Step-by-step solution

Spin two points of the line with the same axis and angle.
Join the two rotated points to get r′.
Check that the distance to the axis is preserved and the line does not change length.
If the line passes through the axis or is parallel to it, the spin may leave it in a special position.

Result: r′ passes through P′(5, 7, 10) with direction (2, 1.6, 3.23).

Spinning a line does not mean spinning “the projection”: points of the line are rotated and joined again.