Ultimate Guide to Vector Intersections and Skew Lines

Introduction

In 2D geometry, lines only have two options: they are parallel, or they cross each other at a single intersection point. However, in the 3D space of HSC Mathematics Extension 2, a third, mind-bending option exists: lines can be skew. Two airplanes flying at different altitudes in different directions will never crash, nor are they flying parallel. In this guide, we use vector algebra to systematically determine exactly how lines in 3D space interact.

Executive Summary

This guide covers the analysis of multiple vector lines:

Parallel Lines: Identifying if two lines travel in the same direction by comparing their direction vectors.
Collinear Points: Proving that three distinct points lie on the exact same straight line.
Intersecting Lines: Setting up simultaneous equations to find the exact $(x, y, z)$ coordinates where two lines cross.
Skew Lines: Proving that two lines are neither parallel nor intersecting.

What is this about?

When you are given two lines, $L_1: \mathbf{r_1} = \mathbf{a_1} + \lambda \mathbf{b_1}$ and $L_2: \mathbf{r_2} = \mathbf{a_2} + \mu \mathbf{b_2}$ (notice we must use different parameters, $\lambda$ and $\mu$ , for different lines!), you must act as a geometric detective.

First, you look at their direction vectors ( $\mathbf{b_1}$ and $\mathbf{b_2}$ ). If they are scalar multiples of each other, the lines are parallel. If they are not parallel, you try to find an intersection. You set the $x, y,$ and $z$ components equal to each other to form three simultaneous equations. Since you only have two unknowns ( $\lambda$ and $\mu$ ), you only need the first two equations to find their values. You then substitute these values into the third equation. If the third equation works, the lines intersect. If the third equation creates a mathematical contradiction (like $5 = 7$ ), the lines miss each other entirely and are skew.

Main Content

1. Parallel and Collinear

Parallel Lines: Two lines are parallel if their direction vectors are scalar multiples. If $\mathbf{b_1} = k \mathbf{b_2}$ for some constant $k$ , the lines are parallel. You do not even need to look at the position vectors.

Collinear Points: Three points $A, B, C$ are collinear (on the same line) if the vector from $A$ to $B$ is parallel to the vector from $A$ to $C$ . You must prove two things:

$\vec{AB} = k \vec{AC}$ (This proves the lines $AB$ and $AC$ are parallel).
They share a common point (Point $A$ ). Therefore, they must be the same continuous line.

2. Finding Intersections

To find where $L_1$ and $L_2$ intersect:

Expand both equations into their $x, y, z$ components.
Equate the $x$ components: $x_1(\lambda) = x_2(\mu)$
Equate the $y$ components: $y_1(\lambda) = y_2(\mu)$
Equate the $z$ components: $z_1(\lambda) = z_2(\mu)$
Solve equations (2) and (3) simultaneously to find the values of $\lambda$ and $\mu$ .
Substitute these values into equation (4) to check for consistency.
If consistent, substitute $\lambda$ back into $L_1$ (or $\mu$ into $L_2$ ) to find the intersection coordinates.

3. Skew Lines

Skew lines only exist in 3D (or higher dimensions). They are lines that are not parallel, but also do not intersect. To prove two lines are skew:

Prove they are not parallel (show $\mathbf{b_1} \neq k \mathbf{b_2}$ ).
Attempt to find an intersection using the simultaneous equations method.
Show that substituting the found $\lambda$ and $\mu$ values into the third equation results in a contradiction (e.g. LHS $\neq$ RHS).
Conclude: "Since the lines are not parallel and do not intersect, they are skew."

Simple Worked Example

Question: Determine whether the following two lines intersect, are parallel, or are skew. If they intersect, find the coordinates of the point of intersection. $L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ $L_2: \mathbf{r} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$

Solution: Step 1: Check for parallel. Direction vectors are $\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ . They are clearly not scalar multiples of each other. Therefore, they are not parallel.

Step 2: Set up simultaneous equations. Equate the $x, y, z$ components:

$1 + 2\lambda = 2 + \mu$ $\implies \mu = 2\lambda - 1$
$2 - \lambda = -1 + \mu$
$-1 + \lambda = 3 + 2\mu$

Step 3: Solve for $\lambda$ and $\mu$ . Substitute equation (1) into equation (2): $2 - \lambda = -1 + (2\lambda - 1)$ $2 - \lambda = 2\lambda - 2$ $4 = 3\lambda \implies \lambda = \frac{4}{3}$

Find $\mu$ : $\mu = 2(\frac{4}{3}) - 1 = \frac{8}{3} - \frac{3}{3} = \frac{5}{3}$

Step 4: Check consistency in the third equation. Substitute $\lambda = \frac{4}{3}$ and $\mu = \frac{5}{3}$ into equation (3): LHS = $-1 + \lambda = -1 + \frac{4}{3} = \frac{1}{3}$ RHS = $3 + 2\mu = 3 + 2(\frac{5}{3}) = \frac{9}{3} + \frac{10}{3} = \frac{19}{3}$

Since LHS $\neq$ RHS ( $\frac{1}{3} \neq \frac{19}{3}$ ), the system of equations has no solution.

Final Answer: The lines are not parallel and do not intersect. Therefore, the lines are skew.

mini-FAQ page

Q: Do I have to use equations 1 and 2 to find the parameters? A: No, you can pick any two equations to solve simultaneously. It is usually best to pick the two equations that look the algebraically simplest. You then check the consistency using the remaining third equation.

Q: What if the question asks to find the distance between two skew lines? A: Finding the shortest distance between skew lines is generally outside the scope of the HSC Mathematics Extension 2 syllabus, unless the question leads you through a very specific geometrical construction using dot products.

Common mistakes to avoid

Using the same parameter for both lines: This is a fatal error! If you write $\mathbf{r_1} = \mathbf{a_1} + \lambda \mathbf{b_1}$ and $\mathbf{r_2} = \mathbf{a_2} + \lambda \mathbf{b_2}$ , you are forcing the lines to only intersect if they happen to reach the crossing point at the exact same "time" $\lambda$ . Two airplanes can cross paths at different times! You MUST use different parameters (e.g. $\lambda$ and $\mu$ ).
Stopping after finding $\lambda$ and $\mu$ : Many students solve the simultaneous equations and forget to substitute the values back into the vector equation to find the actual $(x,y,z)$ coordinate of the intersection.

Practice on Vu's Maths Hub

Proving lines are skew requires rigorous algebraic discipline.

Practice solving complex 3D simultaneous equations in the HSC Vectors Booklet.
Test your collinearity proofs at Vu's Maths Hub.

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