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Vectors – Intersection of two lines

📘 Notes

Vectors — Intersection of Two Lines

In Year 13 vectors, two lines intersect if they pass through the same point. To find whether two lines meet, we compare their vector or parametric equations and solve for the parameters.

This is a standard Cambridge A Level 9709 skill. You need to be able to decide whether two lines intersect, and if they do, find the point of intersection clearly and accurately.

Key definitions and formulae

1. Vector equation of a line

A line in three dimensions can be written as

\[ \mathbf{r}=\mathbf{a}+\lambda \mathbf{b}, \]

where \( \mathbf{a} \) is the position vector of a fixed point on the line, \( \mathbf{b} \) is a direction vector, and \( \lambda \) is a parameter.

2. Two lines

Suppose the two lines are

\[ \mathbf{r}=\mathbf{a}+\lambda \mathbf{b} \] \[ \mathbf{r}=\mathbf{c}+\mu \mathbf{d}. \]

If they intersect, then there is one point that lies on both lines, so the two vector expressions are equal for some values of \( \lambda \) and \( \mu \).

3. Condition for intersection

To test for intersection, set the two position vectors equal:

\[ \mathbf{a}+\lambda \mathbf{b}=\mathbf{c}+\mu \mathbf{d}. \]

Then compare the \(x\)-, \(y\)-, and \(z\)-coordinates to form simultaneous equations in \( \lambda \) and \( \mu \).

4. Possible outcomes

Result Meaning
One consistent pair of parameter values The lines intersect
No consistent solution The lines do not intersect
Direction vectors are multiples The lines may be parallel or the same line

Standard method

  1. Write both lines in vector or parametric form.
  2. Set the two vector equations equal.
  3. Compare corresponding coordinates to form equations in the parameters.
  4. Solve the equations.
  5. Check the third coordinate as well, not just two coordinates.
  6. If the same values of the parameters satisfy all three coordinates, substitute back to find the intersection point.

Worked examples

Example 1: Find the intersection point of two lines

Find the point of intersection of the lines

\[ \mathbf{r}= \begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2\\ -1\\ 4 \end{pmatrix} \] \[ \mathbf{r}= \begin{pmatrix} 5\\ 0\\ 11 \end{pmatrix} + \mu \begin{pmatrix} -1\\ 1\\ -2 \end{pmatrix}. \]

Solution

Set the two expressions equal:

\[ \begin{pmatrix} 1+2\lambda\\ 2-\lambda\\ 3+4\lambda \end{pmatrix} = \begin{pmatrix} 5-\mu\\ \mu\\ 11-2\mu \end{pmatrix}. \]

Compare coordinates:

\[ 1+2\lambda=5-\mu \] \[ 2-\lambda=\mu \] \[ 3+4\lambda=11-2\mu. \]

From \( 2-\lambda=\mu \), substitute into the first equation:

\[ 1+2\lambda=5-(2-\lambda)=3+\lambda \] \[ \lambda=2. \]

Then

\[ \mu=2-\lambda=0. \]

Check in the third equation:

\[ 3+4(2)=11-2(0) \] \[ 11=11. \]

So the lines intersect. Substitute \( \lambda=2 \) into the first line:

\[ \mathbf{r}= \begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} + 2 \begin{pmatrix} 2\\ -1\\ 4 \end{pmatrix} = \begin{pmatrix} 5\\ 0\\ 11 \end{pmatrix}. \]
The point of intersection is \[ \begin{pmatrix} 5\\ 0\\ 11 \end{pmatrix}. \]

Example 2: Show that two lines do not intersect

Determine whether the lines

\[ \mathbf{r}= \begin{pmatrix} 1\\ -2\\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3\\ 1\\ -2 \end{pmatrix} \] \[ \mathbf{r}= \begin{pmatrix} 7\\ 1\\ 0 \end{pmatrix} + \mu \begin{pmatrix} -1\\ 2\\ 1 \end{pmatrix} \]

intersect.

Solution

Set the coordinates equal:

\[ 1+3\lambda=7-\mu \] \[ -2+\lambda=1+2\mu \] \[ 4-2\lambda=\mu. \]

From the third equation,

\[ \mu=4-2\lambda. \]

Substitute into the first equation:

\[ 1+3\lambda=7-(4-2\lambda)=3+2\lambda \] \[ \lambda=2. \]

Then

\[ \mu=4-2(2)=0. \]

Check the second equation:

\[ -2+2=1+2(0) \] \[ 0\neq 1. \]
The equations are inconsistent, so the lines do not intersect.

Example 3: Find the intersection using parametric form

The lines are given by

\[ x=2+t,\qquad y=1-2t,\qquad z=3+t \] \[ x=5-s,\qquad y=-5+4s,\qquad z=6-s. \]

Find whether the lines intersect.

Solution

Compare coordinates:

\[ 2+t=5-s \] \[ 1-2t=-5+4s \] \[ 3+t=6-s. \]

From the first and third equations:

\[ t+s=3 \] \[ t+s=3. \]

So these are consistent. Use \( s=3-t \) in the second equation:

\[ 1-2t=-5+4(3-t) \] \[ 1-2t=7-4t \] \[ 2t=6 \] \[ t=3. \]

Then

\[ s=3-t=0. \]

Substitute into either line:

\[ (x,y,z)=(2+3,\;1-2(3),\;3+3)=(5,-5,6). \]
The lines intersect at \((5,-5,6)\).

Example 4: Parallel lines

Determine the relationship between the lines

\[ \mathbf{r}= \begin{pmatrix} 1\\ 3\\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2\\ -1\\ 4 \end{pmatrix} \] \[ \mathbf{r}= \begin{pmatrix} 5\\ 1\\ 6 \end{pmatrix} + \mu \begin{pmatrix} -4\\ 2\\ -8 \end{pmatrix}. \]

Solution

Compare the direction vectors:

\[ \begin{pmatrix} -4\\ 2\\ -8 \end{pmatrix} =-2 \begin{pmatrix} 2\\ -1\\ 4 \end{pmatrix}. \]

The direction vectors are multiples, so the lines are parallel or identical.

Check whether the point \( \begin{pmatrix} 5\\ 1\\ 6 \end{pmatrix} \) lies on the first line:

\[ 1+2\lambda=5 \Rightarrow \lambda=2 \] \[ 3-\lambda=1 \Rightarrow \lambda=2 \] \[ -2+4\lambda=6 \Rightarrow \lambda=2. \]

All three coordinates give the same value of \( \lambda \), so the point lies on the first line.

The two lines are the same line.

Common mistakes and exam tips

Common mistakes

  • Using only two coordinates and forgetting to check the third.
  • Finding values of the parameters that work in one or two equations but not all three.
  • Thinking equal direction vectors automatically mean the lines intersect.
  • Mixing up the two parameters, for example using \( \lambda \) for both lines.
  • Not distinguishing clearly between intersecting, parallel, and identical lines.

Exam tips

  • Always write both lines in comparable form before starting.
  • After solving for the parameters, substitute back to find the actual point of intersection.
  • If the direction vectors are multiples, test whether one point from one line lies on the other line.
  • State your conclusion clearly: intersect, do not intersect, parallel, or same line.

Summary

\[ \mathbf{a}+\lambda \mathbf{b}=\mathbf{c}+\mu \mathbf{d} \] \[ \text{solve for } \lambda \text{ and } \mu \] \[ \text{check all three coordinates} \]

To find whether two lines intersect, set their vector equations equal, solve for the parameters, and check that the same parameter values satisfy all coordinate equations.