IB Mathematics AA Different types of lines Study Notes
IB Mathematics AA Different types of lines Study Notes
IB Mathematics AA Different types of lines Notes Offer a clear explanation of Different types of lines, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Different types of lines.
Types of Relationships Between Two Lines in Space
Types of Relationships Between Two Lines in Space
When analyzing two lines in 2D or 3D, they can relate to each other in the following ways:
| Characteristics | What You Draw | What You Say | What You Write |
|---|---|---|---|
| Parallel lines never cross and stay the same distance apart. They are coplanar. They have 0 points in common. |  | Line AB is parallel to line CD or line l is parallel to line j | AB ⇄ CD or line l ⇄ line j |
| Intersecting lines pass through the same point. They have one point in common. |  | Lines HG and EF intersect at point I. | HG intersects EF (There is no symbol for intersection of lines.) |
| Perpendicular lines intersect at right angles. They have one point in common. |  | Line LM is perpendicular to line JK. | LM ⊥ JK |
| Coincident lines are the same line. They have an infinite number of points in common. |  | Line NO and line OP are coincident lines. | (There is no symbol for coincident lines.) |
| Skew lines are lines that are non-coplanar and never intersect. They have 0 points in common. |  | Line TS and line QR are skew lines. | TS and QR are skew. (There is no symbol for skew lines.) |
Example
Determine the relationship between the following two lines:
Line 1: \( \mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} \)
Line 2: \( \mathbf{r}_2 = \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} \)
▶️Answer/Explanation
Direction vectors are the same: \( \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} \). So the lines are parallel or coincident.
Check if one point lies on the other line:
Test if \( \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix} \) lies on Line 1:
Solve:
\( 1 + 2\lambda = 3 \Rightarrow \lambda = 1 \)
\( 2 – 1 \cdot 1 = 2 – 1 = 1 \) → does not match 0
Therefore, lines are parallel but not coincident.
Example
Determine the relationship between these lines:
Line 1: \( \mathbf{r}_1 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \)
Line 2: \( \mathbf{r}_2 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} \)
▶️ Answer/Explanation
The direction vectors are not proportional → Not parallel.
Attempt to solve for \( \lambda, \mu \) where lines intersect:
Set:
\( \begin{pmatrix} 0 + \lambda \\ 0 + \lambda \\ 0 + \lambda \end{pmatrix} = \begin{pmatrix} 1 + 0 \mu \\ 0 + 1 \mu \\ 0 – 1 \mu \end{pmatrix} \)
Leads to contradiction → No solution.
Therefore, lines are skew.
Points of Intersection of Lines
Points of Intersection of Lines
To find the point of intersection of two lines:
- Write their vector equations or parametric equations.
- Set the equations equal and solve for the parameters.
- Substitute back to find the point of intersection (if any).
- If no common solution exists, the lines do not intersect (they are either parallel or skew).
Example
Find the point of intersection (if any) of the following lines:
Line 1: \( \mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix} \)
Line 2: \( \mathbf{r}_2 = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ -2 \end{pmatrix} \)
▶️ Answer/Explanation
Set components equal:
\( 1 + 2\lambda = 3 – \mu \) → (1)
\( 2 + \lambda = 0 + 2\mu \) → (2)
\( 3 – \lambda = 4 – 2\mu \) → (3)
From (1): \( 2\lambda + \mu = 2 \)
From (2): \( \lambda – 2\mu = -2 \)
From (3): \( -\lambda + 2\mu = 1 \)
Solving (2) and (3):
Add: \( 0 = -1 \) → Contradiction
No point of intersection. The lines are skew.