Finding a vector equation orthogonal to a plane

Solution 1:

Think it geometrically: if a line's guiding vector is parallel to a plane's normal vector' then it is normal (thus is, orthogonal) to the plane. This is quite intuitive: a plane's normal vector defines the direction orthogonal to it. If a line is parallel to that vector, then it has its direction and, therefore, is orthogonal to the plane.

The solution they provided you is correct, for it goes through the desired point in a direction orthogonal to the plane, this is, the direction of the normal vector.

You can also think geometrically why, if a line's guiding vector is ortohogonal to a plane's normal vector, then it is one of the infinitely many lines that are parallel to the plane

Solution 2:

"Orthogonal" relates to perpendicularity. So you're basically finding a line perpendicular to the plane, which means a single line that's simultaneously and mutually perpendicular to every line on the plane.

What you have in the first step is a normal vector to the plane, which means a vector which is already at right angles to the plane. To get the required line perpendicular to the plane, the normal vector will simply be the direction vector of that line. In other words, you want a line in the direction of (or exactly opposite to) the normal vector of the plane, but not one that's at any other angle (including perpendicular) to the normal vector. That would defeat the purpose - if you calculated a line direction vector perpendicular to that normal vector, you're getting a line that's parallel to a line on the plane. This is not what's required.

So all you need to do is use the normal vector of the plane as the direction vector of the line in the formula (and the given fixed point to "anchor" it in space) and you're done.