Introduction to Lines in 3D Space

In our everyday 2D world, two lines either cross each other (intersect) or run perfectly alongside each other (parallel). However, in the 3D world of Higher Level Mathematics, things get a bit more interesting! Imagine a plane flying high in the sky and a car driving on a road below it. Their paths aren't parallel, yet they will never crash. This is the unique 3D concept of skew lines.

In this chapter, we will learn how to use the vector equations of lines to determine exactly how they relate to one another. This is a vital skill for Topic 3, as it bridges the gap between basic vector algebra and the complex geometry of planes.

The Four Relationships Between Lines

When comparing two lines in 3D space, there are exactly four possibilities:

  1. Coincident Lines: The lines are actually the same line; every point on one is on the other.
  2. Parallel Lines: The lines go in the same direction but never touch.
  3. Intersecting Lines: The lines meet at exactly one unique point.
  4. Skew Lines: The lines are not parallel, but they still never meet (only possible in 3D).

To distinguish between these, we look at two things: their direction vectors and whether they share any points.

Step 1: Checking for Parallelism

Given two lines with equations:
Line 1: \( \mathbf{r}_1 = \mathbf{a}_1 + \lambda \mathbf{b}_1 \)
Line 2: \( \mathbf{r}_2 = \mathbf{a}_2 + \mu \mathbf{b}_2 \)

The first thing we always check is the direction vectors, \( \mathbf{b}_1 \) and \( \mathbf{b}_2 \). If the directions are scalar multiples of each other, the lines are either parallel or coincident.

The Test: If \( \mathbf{b}_1 = k\mathbf{b}_2 \) for some constant \( k \), the lines have the same direction.

How to tell Parallel from Coincident?

If the directions are the same, pick a point from Line 1 (the position vector \( \mathbf{a}_1 \)) and see if it lies on Line 2.
- If \( \mathbf{a}_1 \) is on Line 2, the lines are coincident.
- If \( \mathbf{a}_1 \) is not on Line 2, the lines are parallel.

Quick Tip: "Coincident" is just a fancy math word for "identical."

Step 2: Checking for Intersecting vs. Skew

If the direction vectors are not scalar multiples, the lines are either intersecting or skew. To find out, we try to find a point they share by setting the equations equal to each other.

The "System of Three Equations" Method

Since \( \mathbf{r}_1 = \mathbf{r}_2 \), we compare the components of the two lines:

\( a_{1x} + \lambda b_{1x} = a_{2x} + \mu b_{2x} \)
\( a_{1y} + \lambda b_{1y} = a_{2y} + \mu b_{2y} \)
\( a_{1z} + \lambda b_{1z} = a_{2z} + \mu b_{2z} \)

This gives us a system of three linear equations with two unknowns, \( \lambda \) and \( \mu \).

Step-by-Step Process:

  1. Use any two of the equations to solve for \( \lambda \) and \( \mu \).
  2. Substitute the values you found into the third equation (the "Check" equation).
  3. The Result:
    - If the third equation is consistent (the values work), the lines intersect.
    - If the third equation is inconsistent (e.g., you get \( 5 = 12 \)), the lines are skew.

Did you know? In 2D, non-parallel lines must intersect. Skew lines only exist when we add the third dimension (\( z \)), allowing lines to pass over or under each other.

Finding the Point of Intersection

If you have confirmed that the lines intersect, finding the actual coordinates of that point is easy! Simply take the value of \( \lambda \) (or \( \mu \)) you calculated and substitute it back into its original line equation.

Example: If you found \( \lambda = 2 \) for Line 1, the intersection point \( P \) is found by calculating \( \mathbf{a}_1 + 2\mathbf{b}_1 \).

Summary Flowchart for Exams

Don't worry if this seems like a lot of steps. Just follow this logic flow:

  1. Are the direction vectors multiples?
    • Yes: Check if a point from one is on the other.
      - Point matches \( \rightarrow \) Coincident
      - Point doesn't match \( \rightarrow \) Parallel
    • No: Try to solve the system of equations for \( \lambda \) and \( \mu \).
      - Solution works for all 3 components \( \rightarrow \) Intersecting
      - No consistent solution exists \( \rightarrow \) Skew

Common Mistakes to Avoid

1. Using the same parameter: Always use different symbols for the parameters of two different lines (e.g., \( \lambda \) and \( \mu \)). If you use \( t \) for both, you are accidentally assuming they "reach" the intersection point at the same "time," which isn't necessary for the lines to cross.

2. Forgetting the "Check": In Paper 1 (no calculator), students often solve for \( \lambda \) and \( \mu \) using the \( x \) and \( y \) equations but forget to verify with the \( z \) equation. You must check the third equation to prove the lines aren't skew!

3. Misinterpreting GDC results: In Paper 2, you can use your GDC to solve the system of equations. If the calculator shows "No Solution," and you have already confirmed the directions aren't parallel, then the lines are skew.

Quick Review: Key Takeaways

  • Parallel: \( \mathbf{b}_1 = k\mathbf{b}_2 \), no common points.
  • Coincident: \( \mathbf{b}_1 = k\mathbf{b}_2 \), infinitely many common points.
  • Intersecting: Directions not multiples, \( \exists \) unique \( \lambda, \mu \) satisfying all three component equations.
  • Skew: Directions not multiples, no \( \lambda, \mu \) values satisfy all three component equations.