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

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Amber

Last updated

3 January 2024

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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
- $r = a + λ b$
  - 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
  - $λ$ 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 $y = m x + c$ 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 $r = (\binom{a_{1}}{\begin{matrix} a_{2} \\ a_{3} \end{matrix}}) + λ (\binom{b_{1}}{\begin{matrix} b_{2} \\ b_{3} \end{matrix}})$ the point c with position vector $(\binom{c_{1}}{\begin{matrix} c_{2} \\ c_{3} \end{matrix}})$ is on the line if there exists a value of $λ$ such that
- $(\binom{c_{1}}{\begin{matrix} c_{2} \\ c_{3} \end{matrix}}) = (\binom{a_{1}}{\begin{matrix} a_{2} \\ a_{3} \end{matrix}}) + λ (\binom{b_{1}}{\begin{matrix} b_{2} \\ b_{3} \end{matrix}})$
- This means that there exists a single value of $λ$ that satisfies the three equations:
  - $c_{1} = a_{1} + λ b_{1}$
  - $c_{2} = a_{2} + λ b_{2}$
  - $c_{3} = a_{3} + λ b_{3}$
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 $λ$ exists for all three equations
Solve one of the equations first to find a value of $λ$ that satisfies the first equation and then check that this value also satisfies the other two equations
If the value of $λ$ does not satisfy all three equations, then the point c does not lie on the line

Examiner 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 unit vectors or as column vectors
- Use the form that you prefer, however column vectors is generally easier to work with

Worked example

Find a vector equation of a straight line through the points with position vectors a = 4i – 5k and b = 3i - 3k

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Determine whether the point C with coordinate (2, 0, -1) lies on this line.

3-10-1-ib-aa-hl-vector-equation-of-a-line-we-b

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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 $r = (\binom{x}{\begin{matrix} y \\ z \end{matrix}})$ then $r = a + λ b$ becomes
- $(\binom{x}{\begin{matrix} y \\ z \end{matrix}}) = (\binom{x_{0}}{\begin{matrix} y_{0} \\ z_{0} \end{matrix}}) + λ (\binom{l}{\begin{matrix} m \\ n \end{matrix}})$
  - Where $(\binom{x_{0}}{\begin{matrix} y_{0} \\ z_{0} \end{matrix}})$ is a position vector and $(\binom{l}{\begin{matrix} m \\ n \end{matrix}})$ is a direction vector
- This vector equation can then be split into its three separate component forms:
  - $x = x_{0} + λ l$
  - $y = y_{0} + λ m$
  - $z = z_{0} + λ n$
- These are given in the formula booklet

Worked example

Write the parametric form of the equation of the line which passes through the point (-2, 1, 0) with direction vector $(\binom{3}{\begin{matrix} 1 \\ - 4 \end{matrix}})$ .

3-10-1-ib-aa-hl-parametric-we

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Angle Between Two Lines

How do we find the angle between two lines?

The angle between two lines is equal to the angle between their direction vectors
- It can be found using the scalar product of their direction vectors
Given two lines in the form $r = a_{1} + λ b_{1}$ and $r = a_{2} + λ b_{2}$ use the formula
- $θ = \cos^{- 1} (\frac{b_{1} ∙ b_{2}}{|b_{1}| |b_{2}|})$
If you are given the equations of the lines in a different form or two points on a line you will need to find their direction vectors first
To find the angle ABC the vectors BA and BC would be used, both starting from the point B
The intersection of two lines will always create two angles, an acute one and an obtuse one
- A positive scalar product will result in the acute angle and a negative scalar product will result in the obtuse angle
  - Using the absolute value of the scalar product will always result in the acute angle

Examiner Tip

In your exam read the question carefully to see if you need to find the acute or obtuse angle
- When revising, get into the practice of double checking at the end of a question whether your angle is acute or obtuse and whether this fits the question

Worked example

Find the acute angle, in radians between the two lines defined by the equations:

$l_{1} : a = (\begin{matrix} 2 \\ 0 \\ 3 \end{matrix}) + λ (\begin{matrix} 1 \\ - 4 \\ - 3 \end{matrix})$ and $l_{2} : b = (\begin{matrix} 1 \\ - 4 \\ 3 \end{matrix}) + μ (\begin{matrix} - 3 \\ 2 \\ 5 \end{matrix})$

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