Vector Equations of Lines (DP IB Maths: AA HL)

How do I find the vector equation of a line?

• The formula for finding the vector equation of a line is
  • 
    • Where r is the position vector of any point on the line
    • a is the position vector of a known point on the line
    • b is a direction (displacement) vector
    •  is a scalar
  • This is given in the formula booklet
  • This equation can be used for vectors in both 2- and 3- dimensions
• This formula is similar to a regular equation of a straight line in the form  but with a vector to show both a point on the line and the direction (or gradient) of the line
  • In 2D the gradient can be found from the direction vector
  • In 3D a numerical value for the direction cannot be found, it is given as a vector
• As a could be the position vector of any point on the line and b could be any scalar multiple of the direction vector there are infinite vector equations for a single line
• Given any two points on a line with position vectors a and b the displacement vector can be written as b - a
  • So the formula r = a + λ(b - a) can be used to find the vector equation of the line
  • This is not given in the formula booklet

How do I determine whether a point lies on a line?

• Given the equation of a line the point c with position vector is on the line if there exists a value of such that

• • 
  • This means that there exists a single value of that satisfies the three equations:
    • 
    • 
    • 
• A GDC can be used to solve this system of linear equations for
  • The point only lies on the line if a single value of  exists for all three equations
• Solve one of the equations first to find a value of that satisfies the first equation and then check that this value also satisfies the other two equations
• If the value of  does not satisfy all three equations, then the point c does not lie on the line

How do I find the vector equation of a line in parametric form?

• By considering the three separate components of a vector in the x, y and z directions it is possible to write the vector equation of a line as three separate equations
  • Letting  then  becomes
  • 
    • Where  is a position vector and  is a direction vector
  • This vector equation can then be split into its three separate component forms:
    • 
    • 
    • 
  • These are given in the formula booklet

• The Cartesian equation of a line can be found from the vector equation of a line by
• Finding the vector equation of the line in parametric form
• Eliminating  from the parametric equations
•  can be eliminated by making it the subject of each of the parametric equations
• For example: gives 
• In 2D the cartesian equation of a line is a regular equation of a straight line simply given in the form
•  
• 
• by rearranging 
• In 3D the cartesian equation of a line also includes z and is given in the form
• 
• where 
• This is given in the formula booklet
• If one of your variables does not depend on  then this part can be written as a separate equation
• For example: gives 

How do I find the vector equation of a line given the Cartesian form?

• If you are given the Cartesian equation of a line in the form
  • 
• A vector equation of the line can be found by
  • STEP 1: Set each part of the equation equal to individually
  • STEP 2: Rearrange each of these three equations (or two if working in 2D) to make x, y, and z the subjects
    • This will give you the three parametric equations
    • 
    • 
    • 
  • STEP 3: Write this in the vector form 
  • STEP 4: Set r  to equal 
• If one part of the cartesian equation is given separately and is not in terms of then the corresponding component in the direction vector is equal to zero