Vector Equations of Lines (DP IB Analysis & Approaches (AA)): Revision Note

Equation of a Line in Vector Form

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

Equation of a Line in Parametric Form

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

Equation of a Line in Cartesian Form

• 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