# Write the equation of a line that passes through

Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.

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. Note as well that a vector function can be a function of two or more variables.

## Find the Equation of a Line Given That You Know Two Points it Passes Through

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. 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. Here are some evaluations for our example.

If we do some more evaluations and plot all the points we get the following sketch. In this case we get an ellipse.

- So, consider the following vector function;
- It is important to not come away from this section with the idea that vector functions only graph out lines;
- In this case we get an ellipse;
- However, in those cases the graph may no longer be a curve in space.

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.

- It can be anywhere, a position vector, on the line or off the line, it just needs to be parallel to the line.
- It can be anywhere, a position vector, on the line or off the line, it just needs to be parallel to the line.
- It is important to not come away from this section with the idea that vector functions only graph out lines. If it does give the coordinates of that point.
- That means that any vector that is parallel to the given line must also be parallel to the new line. However, in this case it will.

Vectors give directions and can be three dimensional objects. We now have the following sketch with all these points and vectors on it. It can be anywhere, a position vector, on the line or off the line, it just needs to be parallel to the line.

- There is one more form of the line that we want to look at.
- We know that the new line must be parallel to the line given by the parametric equations in the problem statement. Show Solution To answer this we will first need to write down the equation of the line.
- The vector that the function gives can be a vector in whatever dimension we need it to be.

There are several other forms 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.

Write down all three forms of the equation of the line. However, in this case it will. 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. We could just have easily gone the other way.

## Finding the Equation of a Line

Here is the vector form of the line. Here are the parametric equations of the line. If it does give the coordinates of that point. Show Solution To answer this we will first need to write down the equation of the line. We know a point on the line and just need a parallel vector. We know that the new line must be parallel to the line given by the parametric equations in the problem statement.

That means that any vector that is parallel to the given line must also be parallel to the new line.