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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 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..."
