CIE A Level Maths: Pure 3复习笔记7.3.1 Equation of a Line in Vector Form

Equation of a Line in Vector Form

How do I find the vector equation of a line?

  • You need to know:
    • The position vector of one point on the line
    • direction vector of the line (or the position vector of another point)
  • There are two formulas for getting a vector equation of a line:
    • a + t (b - a)
      • use this formula when you know the position vectors a and b of two points on the line
    • a + t d
      • use this formula when you know the position vector a of a point on the line and a direction vector d
    • Both forms could be compared to the Cartesian equation of a 2D line

How do I determine if a point is on a line?

  • Each different point on the line corresponds to a different value of t
    • For example: if an equation for a line is r = 3i + 2j - + t (i + 2j)
      • the point with coordinates (2, 0, -1) is on the line and corresponds to t = -1
    • However we know that the point with coordinates (-7, 5, 0) is not on this line
      • No value of could make the k component 0

Can two different equations represent the same line?

  • Why do we say direction vector and not the direction vector? Because the magnitude of the vector doesn’t matter; only the direction is important
    • we can multiply any direction vector by a (non-zero) constant and this wouldn’t change the direction
  • Therefore there are an infinite number of options for (a point on the line) and an infinite number of options for the direction vector
  • For Cartesian equations – two equations will represent the same line only if they are multiples of each other

Worked Example



Exam Tip

  • Remember that the vector equation of a line can take many different forms. This means that the answer you derive might look different from the answer in a mark scheme.
  • You can choose whether to write your vector equations of lines using reference unit vectors or as column vectors – use the form that you prefer!
  • If, for example, an exam question uses column vectors, then it is usual to leave the answer in column vectors, but it isn’t essential to do so - you’ll still get the marks!