Intersection of Lines

  2. Geometry & Trigonometry
  3. Intersection of Lines
    4. 30'

This page looks at the intersections of lines in 2 and 3 dimensions. In 2D, lines intersect or are parallel. In 3D, it is slightly more complicated, they can intersect or be parallel or be skew (not parallel and not intersect!). We will also look at applications of this with moving objects (Kinematics) in which boats and planes can cross paths and collide.

Key Concepts

On this page, you should learn about

  • coincident, parallel, intersecting and skew lines
  • the point of intersection of two lines
 

Essentials

The following videos will help you understand all the concepts from this page

Intersection of 2 Lines

In the following video, we are going to look at the intersection of two lines. We will consider the following example

Find the coordinates of the point of intersection of the following two lines

\(\textbf{r}_1=\left( \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} 2 \\ -3 \\ 1 \end{matrix} \right) \)

\(\textbf{r}_2=\left( \begin{matrix} 5 \\ 1 \\ 0 \end{matrix} \right) +\mu\left( \begin{matrix} 1 \\ 2 \\ -1 \end{matrix} \right) \)

Notes from the video

 

Kinematics and Collisions

In the following video we are going to look we will try and gain a conceptual understanding of velocity vectors. One of the key ideas of this topic is to decide if objects collide. It is not enough to consider if their paths cross. We need to think about whether they occupy the same position at the same moment in time.

To get you started, you might like to play this game to give you an idea about what is going on. Try to hit the submarine with the torpedo!

Now let's consider the example below:

A submarine is initially positioned at (0, 5) travels with velocity \(\left( \begin{matrix} 4 \\ -3 \end{matrix} \right) \\ \)ms-1 .

One second later a torpedo is fired from (3, 0) with velocity \(\left( \begin{matrix} 5 \\ 1 \end{matrix} \right) \\ \)ms-1 .

Does the torpedo manage to shoot the submarine?

​​ Notes from the video

 
 

Summary

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Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

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Exam-style Questions

Question 1

A line L1 passes through the points P(-13,-6,1) and Q(3,2,-3).

A second line L2 has equation \(\textbf{r}=\left( \begin{matrix} 9\\12 \\ 2 \end{matrix} \right) +s\left( \begin{matrix} -3 \\ 2\\4 \end{matrix} \right) \)

  1. Show that \(\overrightarrow{PQ}=\left( \begin{matrix} 16 \\ 8\\-4 \end{matrix} \right) \)
  2. Hence, write down the equation L1 in the form \(\textbf{r}=\textbf{a}+t \textbf{b}\)
  3. The lines L1 and L2 intersect at the point R. Find the coordinates of R.

Hint

Full Solution

 

Question 2

The diagram shows quadrilateral ABCD with vertices A(6,0) , B(3,5) , C(-10,4) and D(1,-3).
  1. Find \(\overrightarrow{AC}\)
  2. Show that \(\overrightarrow{BD}\) is perpendicular to \(\overrightarrow{AC}\)
  3. Write down the equation of the line (AC) in the form
  4. Write down the equation of the line (BD)
  5. The lines (AC) and (BD) intersect at E. Find the coordinates of E

Hint

Full Solution

 

Question 3

Two boats A and B, move so that a time t hours, their positions, in kilometres, are given by

​\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)​

\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)

  1. Find the position where the two boats cross.
  2. Show that the boats do not collide.

Hint

Full Solution

Two boats A and B, move so that a time t hours, their positions, in kilometres, are given by

​\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)​

\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)

  1. Find the position where the two boats cross.
  2. Show that the boats do not collide.

Hint

Full Solution

Two boats A and B, move so that a time t hours, their positions, in kilometres, are given by

​\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)​

\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)

  1. Find the position where the two boats cross.
  2. Show that the boats do not collide.

Hint

Full Solution

Question 4

The line L1 has equation ​\(\textbf{r}=\left( \begin{matrix} 2\\-1 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} -1 \\ -2\\k \end{matrix} \right) \)​

The line L2 has equation \(\textbf{r}=\left( \begin{matrix} 2\\1 \\ -3 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -1\\-1 \end{matrix} \right) \)

  1. The point A(3,1,-1) lies on the line L1. Show that k = 4.
  2. Show that the lines and L1 are L2 perpendicular.
  3. Show that the lines and L1 do not L2 intersect.
  4. The point B lies on the line The point C has coordinates (2,1,-3). ABC forms an isosceles triangles with AC=BC. Find the coordinates of B.

Hint

Full Solution

 
 
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