Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Just like we have the formula $y=mx+b$ for $\mathbb{R}^{2}$, what would be a formula for $\mathbb{R}^{3}$? Thanks.

share|cite|improve this question
$$ (a_0 + a_1 t, b_0 + b_1 t,c_0 + c_1 t) $$ where $a_1, b_1, c_1$ are not all $0.$ – Will Jagy May 28 '13 at 3:39
Was there something in particular that was not clear from the many resources available upon Googling "3d line equation"? – Zev Chonoles May 28 '13 at 3:39
up vote 5 down vote accepted

You can describe a line in space as the intersection of two planes. Thus, $$\{(x,y,z)\in{\mathbb R}^3: a_1x+b_1y+c_1z=d_1 \text{ and } a_2x+b_2y+c_2z=d_2\}.$$ Alternatively, you can use vector notation to describe it as $$\vec{p}(t) = \vec{p}_0 + \vec{d}t.$$

I used this relationship to generate this picture:

enter image description here

This is largely a topic that you will learn about in a third semester calculus course, at least in the states.

share|cite|improve this answer
One representation uses 8 numbers and the other uses 6. Is there a representation that uses fewer than 6? – Samuel Danielson Jul 9 at 3:15

Here are three ways to describe the formula of a line in $3$ dimensions. Let's assume the line $L$ passes through the point $(x_0,y_0,z_0)$ and is traveling in the direction $(a,b,c)$.

Vector Form


Here $t$ is a parameter describing a particular point on the line $L$.

Parametric Form


These are basically the equations that result from the three components of vector form.

Symmetric Form


Here we assume $a,b,$ and $c$ are all nonzero. All we've done is solve the parametric equations for $t$ and set them all equal.

share|cite|improve this answer
In my opinion, the symmetric form is the most useless one. – Hawk May 28 '13 at 4:05

I am giving you an example. Let $A(-2,0,1),~~B(4,5,3)$ be two points in $\mathbb R^3$. And let $C$ be the end point for the vector which is drawn from the orrigin. In addition, we assume that this vector has the same direction as the vector $AB$. So we have its coordinates is $(4,5,3)-(-2,0,1)=(6,5,2)$. Therefore the equation of the line passing through $A$ and $B$ is $$L_{AB}: x=(-2,0,1)+t(6,5,2)$$

share|cite|improve this answer
Please advise my friend if you have the time.Thank you.… – Software May 28 '13 at 16:58
@BabakS.: Nice answer + 1, and congratulations on doing 1000 edit reviews - I know how hard those are to do my friend! – Amzoti May 28 '13 at 19:45
@Amzoti: Thanks so much. Yes indeed it was. Huuuh :-) – Babak S. May 28 '13 at 19:46
Hello, dear friend! I hope your students did well on the exam you gave them! – amWhy May 29 '13 at 0:28
@amWhy: I am red-penciling their jobs. They were not so bad. – Babak S. May 29 '13 at 4:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.