When I learnt linear algebra, I found two similar titles using different preposition. Here they are:

Vector Form for the Equation of a Line. // of a line
Parametric Equations for a Line         // for a line

What's the difference between these two? BTW, I'm always confused about using preposition, could someone shed some light on it, or just give me some suggestion?

  • It's unclear without knowing whether there are mathematical differences between the two, which is not discussed in your post. – Katherine Lockwood Jan 12 '17 at 2:29

It may be that you're confusing these because you're focusing on the word 'Line' rather than breaking up the prepositional phrases. The first example has two prepositional phrases, the second example only has one. In both examples, the format "A FOR B" is used, just that B in the first example contains another prepositional phrase in the format "B OF C". The fact that both examples happen to use the words 'Equation' and 'Line' are irrelevant.

Vector Form FOR the Equation OF a Line

The Vector Form is FOR the Equation. The Equation is OF a Line. In other words the Vector Form is FOR (the equation of a line). The FOR shows that the Vector Form is used by whatever comes in the prepositional phrase, in this case (the equation of a line). You could replace (the equation of a line) with anything else, such as Vector Form FOR (a matrix).

The second prepositional phrase explains what kind of Equation it is. In this case, it is the Equation OF (a line).

Parametric Equation FOR a Line

The Equation is FOR a Line. Following the same format as above: Parametric Equation FOR (a Line). Again (a Line) could be substituted for anything else, such as Parametric Equation FOR (a Circle).

Vector Form for the Equation of a Line

Equation already belongs to line

Parametric Equations for a Line

Assigning/deriving parametric equations for a line. Once you have them, you can say, "parametric equations of the line are..."

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