In this explainer, we will learn how to find the parametric equations of straight lines in space.
Let us remind ourselves first about the different forms of equations of a straight line in the -plane (i.e., in two dimensions, 2D). The form gives us the slope and the -intercept . In other words, the direction vector of the line is and the line goes through the point .
From the form , we know that the direction vector of the line is and that the point of coordinates is on the line.
And, finally, when the equation of the line is in the form , we find that the direction vector of the line is (or or , etc.) and that the line passes through the point , where .
Whatever the form of the equation, the two key pieces of information that define a line are its direction vector and one of its points. Let us see how the reasoning works in 2D before moving on to three dimensions (3D).
If we have a line of direction vector that passes through two points and , then vector with components is collinear with vector . In other words, is a scalar multiple of . Therefore, we have where is a real number.
From the above equation, we find that
and
Substituting (1) into (2), we find that
Therefore, the slope of the line that goes through the points and is given by
Let us now consider a line in space whose direction vector is and that goes through the point . For any other point that lies on the line, and are collinear; therefore, , where is a real number. The figure below illustrates this vector equation, with being a point of the line such that .
We can find the same equation by rewriting as and, using the vector position for , and that for , , we find that ; that is, ; it is the equation of the line in vector form.
Considering the components of and , since , we find that is,
If we let vary from to , the above three equations describe the coordinates of all the points on the line. They describe the coordinates of point when .
This set of three equations is called the parametric equations of a straight line in space. Since there are infinitely many points that lie on the line and any vector is a direction vector of the line, there is not a unique set of parametric equations. However, they will all describe the coordinates of all the points on the line (as varies from to , there is no limitation!), and they all define unambiguously the same line.
Definition: Parametric Equations of a Straight Line in Space
The parametric equations of a line in space are a nonunique set of three equations of the form where are the coordinates of a point that lies on the line, is a direction vector of the line, and is a real number (the parameter) that varies from to .
Let us look at the first example.
Example 1: Finding the Parametric Equation of a Line Given a Point and Its Direction Vector
Give the parametric equation of the line on point , with direction vector .
Answer
The parametric equations of a line are of the form where are the coordinates of a point that lies on the line, is a direction vector of the line, and is a real number (the parameter) that varies from to .
Here, we know that lies on the line; therefore, we will substitute these coordinates into the equation for ; the components of the direction vector are , so we substitute these for . We find
This is a parametric equation of the line on point , with direction vector .
It is worth noting that this set of equations that defines the line on point , with direction vector is not unique. We could take, for instance, as the direction vector of the line and find the parametric equations
We could also find the coordinates of another point that lies on the line by choosing a value for . With, for example, , we find, using our first set of parametric equations, that the point of coordinates lies on the line. Using these coordinates gives another set of parametric equations; namely,
Let us now find the parametric equations of a line that passes through two given points.
Example 2: Finding the Parametric Equation of a Line Given Two Points
Write the equation of the straight line passing through the points and in parametric form.
- , , , for
- , , , for
- , , , for
- , , , for
- , , , for
Answer
The parametric equations of a line are of the form where are the coordinates of a point that lies on the line, is a direction vector of the line, and is a real number (the parameter) that varies from to .
Here, we are given two points that lie on the line. To find the components of a direction vector, we simply need to find the components of, for example, . They are .
We could now substitute the coordinates of either or for and the components (or, e.g., ) for .
Since we have here a limited choice of answers, we can start by identifying the components of the direction vectors used in each set of equationsβthey are given by the coefficients of in each equation:
We see that only answer B uses a correct direction vector. Let us now check that the coordinates used in the equations in answer B are correctβthese are the constants in each equation, that is, the coordinates obtained when . We find , that is, the coordinates of . Hence, answer B is a correct set of parametric equations.
Note that the coordinates of used in the equations could be those of neither nor , but the equations could still be correct. In this case, having found a direction vector of the line (), we would have to check that the point of coordinates lies on the line. For this, we need to verify that the vector (or ) is collinear with , that is, verify that
We see that it is equivalent to checking that there exists a value such that the coordinates of verify the parametric equations:
How can we find parametric equations of a line from its Cartesian equations? Remember that the Cartesian equations of a line in space are in the form where are the coordinates of a point that lies on the line, and is a direction vector of the line where , , and are all nonzero real numbers. This form of equations is closely related to the set of parametric equations since they simply give the three expressions for that we get from each of the parametric equations: is equivalent to
When one component of the direction vector is zero, it means that the corresponding coordinate of all the points lying on the line is constant. For instance, if the direction vector of a line is and the point lies on the line, then the parametric equations of the line are
The Cartesian equations of the line are then
The line is perpendicular to the -axis and is in a plane parallel to the -plane.
If the direction vector is unidimensional, that is, two of its components are zero, then the line is parallel to one of the axes. For instance, if the line is parallel to the -axis and passes through the point , its parametric equations are and its Cartesian equations would be , . Compare this equation in 3D with the equation of a line in 2D that is parallel to the -axis: . The value of is the same for all points and nothing is said about the -coordinate because it can take up any value; the set of the -coordinates of all points lying on the line is . It is the meaning of as well when varies from to . Note that we could take as well βall values of would describe as well when varies from to .
Let us practice converting Cartesian equations of a line into parametric equations.
Example 3: Finding the Parametric Equation of a Line Given Its Cartesian Equations
Find the parametric equations of the straight line .
Answer
The Cartesian equations given here have been slightly rearranged from the standard form . However, it does not matter, as we simply need to write that to find a set of parametric equations by rearranging each equation. We find
Note that there is not a unique set of parametric equations as there is not a unique set of Cartesian equations of the same line either. Here, for instance, as we have found that the direction vector is , we could have chosen to take to write our parametric equations. This is equivalent to taking as parameter instead of , that is, having as Cartesian equations .
We will now look at a final example where we need to find the parametric equations of the diagonal of a cube.
Example 4: Finding the Parametric Equation of a Line in Two Steps
A cube of side length 3 sits with a vertex at the origin and three sides along the positive axes. Find the parametric equations of the main diagonal from the origin.
Answer
Let us start by drawing a diagram of the cube.
The main diagonal of the cube goes from the origin of coordinates to the furthest vertex from that at the origin, namely, the point , since the side of the cube is 3 length units.
The line that contains the diagonal has therefore as direction vector the vector that goes from the origin to the point , that is, the vector of components
The parametric equations of a line are of the form where are the coordinates of a point that lies on the line, is a direction vector of the line, and is a real number (the parameter) that varies from to .
Taking here the origin for the point that lies on the line, we find
Note that we could have taken the point instead, leading to
Or we could have taken as direction vector, leading to the simplest equations
In short, any point on this line has equal -, -, and -coordinates.
In the previous example, we could have limited the possible values for the parameter t to describe only the diagonal of the cube, that is, the line segment that goes from to . With the equations , , , it means that . The range of depends on the equations used. If we take the parametric equations , , , then the range makes these equations describe the diagonal of the cube.
Key Points
- The parametric equations of a line are a nonunique set of three equations of the form where are the coordinates of a point that lies on the line, is a direction vector of the line, and is a real number (the parameter) that varies from to .
- When , , and are all nonzero real numbers, the parametric equations of a straight line can be derived from its Cartesian equations by writing and rearranging each of the three resulting equations.
- When one component of the direction vector is zero, it means that the corresponding coordinates of all the points lying on the line are constant. For instance, if the direction vector of a line is and the point lies on the line, then the parametric equations of the line are and the Cartesian equations of the line are then
- If two components of the direction vector are zero, the line is parallel to one axis, meaning that only one coordinate varies, while the other two are fixed. For instance, the parametric equations of a line parallel to the -axis and passing through point are and its Cartesian equations are , .
