6.1.2 Pairs of Lines in 3D

Parallel and Skew Lines

Parallel Lines: Two lines are parallel if their direction vectors are scalar multiples:

\mathbf{b}_1 = k \mathbf{b}_2, \quad k \in \mathbb{R}

Skew Lines: Skew lines are not parallel and do not intersect. They exist in 3D space without lying in the same plane.

Two lines intersect if there exists a common point.

Let:

\mathbf{r}_1 = \mathbf{a}_1 + \lambda \mathbf{b}_1, \quad \mathbf{r}_2 = \mathbf{a}_2 + \mu \mathbf{b}_2

At the intersection, $\mathbf{r}_1 = \mathbf{r}_2$ , giving three equations:

a_{1i} + \lambda b_{1i} = a_{2i} + \mu b_{2i}, \quad \text{for } i = 1, 2, 3

Solve these simultaneously to find $\lambda$ and $\mu$ .

If a consistent solution exists, substitute $\lambda$ or $\mu$ back to find the intersection point.

lightbulbExample

Example: Find the Intersection of Two Lines

Find the intersection of:

\mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}, \quad \mathbf{r}_2 = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ 1 \\ 3 \end{pmatrix}

Step 1: Set equations equal:

\begin{pmatrix} 1 + \lambda \\ 2 - \lambda \\ 3 + 2\lambda \end{pmatrix} = \begin{pmatrix} 2 - 2\mu \\ \mu \\ 1 + 3\mu \end{pmatrix}

Step 2**: Write component equations:**

Step 3**: Solve simultaneously:**

From: $2 - \lambda = \mu: \mu = 2 - \lambda$

Substitute $\mu = 2 - \lambda$ into $1 + \lambda = 2 - 2\mu$ :

1 + \lambda = 2 - 2(2-\lambda)

Simplify:

1 + \lambda = 2 - 4 + 2\lambda \implies 1 = -2 + \lambda \implies \lambda = 3

Substitute $\lambda = 3$ into $\mu = 2 - \lambda$

\mu = 2 - 3 = -1.

Step 4**: Find the intersection point:**

Substitute $\lambda = 3$ into $\mathbf{r}_1$ :

\mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + 3 \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} = \begin{pmatrix} 4 \\ -1 \\ 9 \end{pmatrix}

Result:

The lines intersect at (4, -1, 9)

infoNote

Confusing the definitions of parallel and skew lines: Parallel lines have direction vectors that are scalar multiples, while skew lines do not intersect and are not parallel.
Errors in simultaneous equations when finding intersections: Mismanagement of algebra when solving $\lambda$ and $\mu$ can lead to incorrect results.
Assuming lines must intersect: Always check the solution for consistency; lines may be skew and not intersect.
Misinterpreting vector components: Ensure proper alignment of vector components when forming equations from vector representations of lines.
Omitting back-substitution to confirm intersection points: Forgetting to substitute $\lambda$ or $\mu$ into the original equations may result in unverified or incorrect solutions.

infoNote

Parallel Lines: Two lines $\mathbf{r}_1 = \mathbf{a}_1 + \lambda \mathbf{b}_1$ and $\mathbf{r}_2 = \mathbf{a}_2 + \mu \mathbf{b}_2$ are parallel if:

\mathbf{b}_1 = k \mathbf{b}_2, \quad k \in \mathbb{R}

\mathbf{a}_1 + \lambda \mathbf{b}_1 = \mathbf{a}_2 + \mu \mathbf{b}_2

Solve for $\lambda$ and $\mu$ by forming three simultaneous equations for the $x, y$ , and $z$ components.

Skew Lines: Skew lines do not satisfy the conditions for parallelism or intersection.
Intersection Point Verification: After finding $\lambda$ and $\mu$ , substitute back into $\mathbf{r}_1$ or $\mathbf{r}_2$ to determine the intersection point.