Equation of a Line

You should already be familiar with the equation of a straight line in Cartesian form in 2 dimensions, y = ax + b. When we move into 3 dimensions, the Cartesian form becomes a little more awkward. Don't worry, vectors are here to help us out! Once you understand the vector equation of a line, it is really useful for solving all sorts of problems with angles and intersections.


Key Concepts

On this page, you should learn about

  • vector equations of lines in two and three dimensions in the three different forms
    • vector form
    • parametric form
    • cartesian form

Essentials

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

Vector Equation of a Line

In the following video we are going to look at the equation of a straight line in vector form and gain a strong conceptual understanding of what this formula means

\(\textbf{ r }\ =\ \overrightarrow { OA } +\lambda \textbf{b}\)

You are probably used to Cartesian form of the equation of a straight line (y = mx + c) and perhaps are wondering why you need vectors to describe a straight line. There are number of answers to this

  1. The Cartesian form is a bit messy when used in 3D.
  2. Finding intersections and the angles which lines meet is easier in vector form.
  3. Describing motion is really helpful using velocity vectors.

Let's start by looking at an example in 2D and then we can move into 3D

Notes from the video

Converting between the different Forms

In the following video we are going to look at the three different forms of the equation of a straight line. In particular, we are going to look at how we can convert from one form to another.

\(\textbf{r}=\left( \begin{matrix} 1 \\ -2 \\ 3 \end{matrix} \right) +\lambda \left( \begin{matrix} -1 \\ 3 \\ 4 \end{matrix} \right) \)

Vector Form

\(x=1-\lambda \\ y=-2+3\lambda \\ z=3+4\lambda \)

Parametric Form

\(\frac { x-1 }{ -1 } =\frac { y-(-2) }{ 3 } =\frac { z-3 }{ 4 } \)

Cartesian Form

Notes from the video

Example - Equidistant Points

In the following video we are going to look at a lovely application of the equation of the straight line to find points equidistant from another point. We don't necessarily have to do it in this way, but it might help us really understand what the vector equation of a straight line means.

Here is the example

A(3,-1,2) and B(6,-7,-7) lie on a straight line L. C also lies on the straight line L. Find the coordinates of the point C given that \(\left| \overrightarrow { AC } \right| =\left| \overrightarrow { AB } \right| \).

Notes from the video

Velocity Vectors

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

Print from here

Test Yourself

Here is a quiz from about equations of lines in vector form


START QUIZ!

The following quiz tests your understanding of converting between the different forms of a straight line (vector, parametric and Cartesian).


START QUIZ!

Exam-style Questions

Question 1

A line L passes through the points A(1,-1,3) and B(3,-4,4)

Point C (x,y,1) also lies on the line L. Find x and y.

Hint

Full Solution

 

Question 2

A line L passes through the points A(0,2,-4) and B(3,-3,2)

Point C also lies on the line L. Find the coordinates of C given that \(\left| \overrightarrow { AC } \right| =\left| \overrightarrow { AB } \right| \)

Hint

Full Solution

 

Question 3

A line L passes through the points A(0,2,-4) and B(3,-3,2)

Point C also lies on the line L. Find the possible coordinates of C given that \(\left| \overrightarrow { AC } \right| =2\left| \overrightarrow { AB } \right| \)

Hint

Full Solution

 

MY PROGRESS

How much of Equation of a Line have you understood?