I am confused about the precise meaning of the phrase "passing through" in geometry. I'm studying vectors and coordinate geometry in 3-space and this phrase is used a lot in problems such as, "plane passing through a point/line/etc".
For example, if a plane passes through a line, does that mean that every point on the line is also a point of the plane?
If the plane "passes through" a line then every point of the line is intersected by the plane, so yes, the line is then contained entirely in the plane. Generally speaking you will only see the phrase used as a higher-dimensional object "passing through" a lower-dimensional object. If you did see it the other way round: "the line passing through the plane" then more context will be needed to understand if the line is part of the plane or just intersecting it at a single point.
It is generally better, in your own work, to be exact about these things (things should intersect or be contained in), but "passing through" was common terminology a hundred years ago when textbooks were more descriptive and less mathematical (I'm thinking of authors like George Salmon, for example) and still has currency today.