To determine the point on the [tex]\( y \)[/tex]-axis which is closest to the point [tex]\((-3.6, 0)\)[/tex] and lies on a line perpendicular to the line passing through [tex]\((-3.6, 0)\)[/tex], we can proceed as follows:
1. Recognize that the point on the [tex]\( y \)[/tex]-axis we are looking for must have the [tex]\( x \)[/tex]-coordinate equal to 0, making it of the form [tex]\((0, y)\)[/tex].
2. The problem involves finding the perpendicular line from the given point [tex]\((-3.6, 0)\)[/tex] to the [tex]\( y \)[/tex]-axis. Perpendicular lines to the [tex]\( y \)[/tex]-axis are horizontal lines.
3. The horizontal line passing through the point [tex]\((-3.6, 0)\)[/tex] contains all points with the same [tex]\( y \)[/tex]-coordinate as [tex]\((-3.6, 0)\)[/tex], which is [tex]\( y = 0 \)[/tex].
4. The intersection of this horizontal line with the [tex]\( y \)[/tex]-axis occurs exactly at the origin [tex]\((0, 0)\)[/tex].
Therefore, the point on the [tex]\( y \)[/tex]-axis that is closest to the point [tex]\((-3.6, 0)\)[/tex] and lies on a line perpendicular to a vertical line passing through [tex]\((-3.6, 0)\)[/tex] is [tex]\((0, 0)\)[/tex].
