# Parametric equation of a line in 3d

Use Mozilla Firefox or Safari instead to view these pages. There are several ways to produce graphs of functions. This page will not show all of them; it will show the way that is the most customizable and can be easily extended to more complicated settings like parametric equations, 3D, etc. This is also a learn-by-example page.

I do not explain every command or show everything. It is up to you to experiment and look up these commands to better understand them. To plot vector functions or parametric equations, you follow the same idea as in plotting 2D functions, setting up your domain for t. Then you establish x, y and z if applicable according to the equations, then plot using the plot x,y for 2D or the plot3 x,y,z for 3D command.

Here are some parametric equations that you may have seen in your calculus text Stewart, Chapter Notice the plot above is extremely jagged. That is because our domain is too "rough" - there aren't enough points plotted.

This is a case where we want to increase number of elements in our x vector.

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This example shows how to plot more than one on the same graph and how to change some features of the graph. This is the same graph as the one above, but getting a different view using view command in order to see that the line segment is tangent to the curve.

This command may take some reading, some practice and even a few runs with different settings to get the correct view. Many of the following work for both the 2D and 3D figures. If we wanted just a plot of points rather than connecting them with a line, we specify a "marker style". Common ones are.

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Using the following LineWidth setting changes the thickness of the line. The default width is 0. Notice that you can barely see the point in the graph above. Default value for MarkerSize is 6. You can also change the way the marker looks by also specifying a different LineSize default is 0. For more information probably more than you want to knowgo to MathWorks documentation. Another way is to use the axis command to set all limits at once with the order xmin, xmax, ymin, ymax, zmin, zmax.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. One possible parametrization of the line could be given like this:. As these are always linear equations, you want to find a particular solution and a solution to the homogenous equation.

What does this mean? To find a particular solution, you want to solve for a single point, that fulfills your equations. Next you want to find a solution to the homogenous equation.

This means removing all constants from your equation and solving that. Here you want to find all solution. Adding a homogenous solution to your particular solution will keep the equation valid.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Parametric form of a line in 3D Ask Question. Asked 1 year, 2 months ago. Active 1 year, 2 months ago. Viewed times. Tarek Chafei Tarek Chafei 87 7 7 bronze badges. Active Oldest Votes. ### Math Insight

Math Linear algebra Vectors and spaces Vectors. Vector intro for linear algebra. Multiplying a vector by a scalar. Practice: Scalar multiplication. Parametric representations of lines. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Everything we've been doing in linear algebra so far, you might be thinking, it's kind of a more painful way of doing things that you already knew how to do.

But in this video I hope to show you something that you're going to do in linear algebra that you've never done before, and that it would have been very hard to do had you not been exposed to these videos.

Well I'm going to start with, once again, a different way of doing something you already know how to do.

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So let me just define some vector here, instead of making them bold, I'll just draw it with the arrow on top. I'm going to define my vector to be-- I can do with the arrow on top or I can just make it super bold. I'm just going to define my vectors, it's going to be a vector in R2. Let's just say that my vector is the vector 2, 1. If I were to draw it in standard position, it looks like this. You go two to the right, up one, like that. That's my, right there, that is my vector v.

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Now, if I were to ask you, what are all of the possible vectors I can create?So, before we get into the equations of lines we first need to briefly look at vector functions. At this point all that we need to worry about is notational issues and how they can be used to give the equation of a curve. The best way to get an idea of what a vector function is and what its graph looks like is to look at an example.

So, consider the following vector function. A vector function is a function that takes one or more variables, one in this case, and returns a vector. Note as well that a vector function can be a function of two or more variables. However, in those cases the graph may no longer be a curve in space. The vector that the function gives can be a vector in whatever dimension we need it to be.

Now, we want to determine the graph of the vector function above. So, to get the graph of a vector function all we need to do is plug in some values of the variable and then plot the point that corresponds to each position vector we get out of the function and play connect the dots.

## Parametric representations of lines

Here are some evaluations for our example. So, each of these are position vectors representing points on the graph of our vector function.

The points.

Parametric Equations of Line Passing Through a Point

In this case we get an ellipse. It is important to not come away from this section with the idea that vector functions only graph out lines. Okay, we now need to move into the actual topic of this section. In this case we will need to acknowledge that a line can have a three dimensional slope. So, we need something that will allow us to describe a direction that is potentially in three dimensions. We already have a quantity that will do this for us. Vectors give directions and can be three dimensional objects. It can be anywhere, a position vector, on the line or off the line, it just needs to be parallel to the line.

This is called the vector form of the equation of a line. There are several other forms of the equation of a line. This set of equations is called the parametric form of the equation of a line. The only difference is that we are now working in three dimensions instead of two dimensions.

In the vector form of the line we get a position vector for the point and in the parametric form we get the actual coordinates of the point. There is one more form of the line that we want to look at. Doing this gives the following. However, in this case it will.

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Since these two points are on the line the vector between them will also lie on the line and will hence be parallel to the line. Note that the order of the points was chosen to reduce the number of minus signs in the vector.

We could just have easily gone the other way. Here is the vector form of the line.This lesson will involve a few examples and applications of the parametric form of a straight line.

### Equation of a line passing through two points in 3d

Solution First of all, let me illustrate a method of finding out the required coordinates without using the parametric form. Therefore, we have two points which satisfy the given conditions: 5, 4 and -3, Note that A is the midpoint of these two points.

It has to be! Since the point A lies on the given line, we can use the parametric form of its equation.

What now? The difference here is that the point, from which the distance is given, does not lie on the given line. Geometrically, this means there will be two such points on the given line. Also, there will be a case when there is exactly one point satisfying the condition, and another in which no such point would exist. Try to figure that out yourself. Note that you could have used the first method as was used in the solution of the previous problem.

Solution Another similar problem. This is too much for you to digest at one go. You must be logged in to post a comment. Let the coordinates of the required point be x 1 ,y 1. We can still use the parametric equation. One more. Leave a comment Cancel reply You must be logged in to post a comment.He still trains and competes occasionally, despite his busy schedule. To unlock all 5, videos, start your free trial.

In the 3D coordinate systemlines can be described using vector equations or parametric equations. Lines in 3D have equations similar to lines in 2D, and can be found given two points on the line. In order to understand lines in 3Done should understand how to parameterize a line in 2D and write the vector equation of a line. Perpendicular, parallel and skew lines are important cases that arise from lines in 3D.

One of the great things about the vector equation of a line is that the same equation applies in 2 dimensions as in 3 dimensions so that's our goal today is to come up with equations for lines and 3 dimensions. So all we need is one position vector and a direction vector and we can get the equation of a line. Let's see an example, find the vector and parametric equations of the line through point a 2, 3, 1 and b 5, 4, 6 and I've drawn the line here.

I have to define a point p this is just going to be a random point on the line and this will give me my r my vector r and I'm going to make this vector r0 you can actually go with either a or b but I'm going to go with point a.

So point a is going to give me my r0, so remember the equations r equals r0 plus t times v. So I need to come up with the components for all of these vectors, now vector r is going to have components the same as the coordinates of this point x, y and z and r0 will have components the same as point a.

So r0 is 2, 3, 1 but we need a direction vector, so what I'm going to do is I'm going to define vector ab as the direction vector so it's nice to have that second point you really need it.

And so v is going to be 3, 1, and 5 that's our direction vector, and then our equation becomes x, y, z equals and r0 was 2, 3, 1 plus t times the direction vector 3, 1, 5.

This is the vector equation for our line, now what the parametric equations? Well remember just like in 2 dimensions you can find the parametric equations by isolating each component. These are the parametric equations they come right out of the vector equation. So when you're asked to find both the vector equation and the parametric equation it's really not that much harder just to go ahead and write the parametric equations down.

So that's it, if you know 2 points on a line you can have with the vector n parametric equations of the line. Previous Unit Advanced Trigonometry. Norm Prokup. Thank you for watching the video.He still trains and competes occasionally, despite his busy schedule. To unlock all 5, videos, start your free trial. Now recall that the vector equation is of the form r equals r0 plus t times v.

And recall that r0 is a position vector for one of the points that you know, is on the line, and v is the direction vector of the line. This is the vector equation. How do we get the parametric equation? We just look at the components, so we get x equals 4 plus 2t.

These three are the parametric equations for my line. Now it turns out that there is one more form for the equation of a line in space. You get it by eliminating the parameter. So I take my parametric equations; x equals 4 plus 2t, y equals -1 plus 3t, and z equals 2 plus t.

I eliminate the parameter.