# Equations of Planes from Symmetric Equation of a Line

• Ocata
In summary, the conversation discusses the process of converting a vector equation into symmetric equations and then splitting them into two separate equations representing planes in standard form. The method of cross multiplication is used to obtain the two equations, which can then be used to find the intersection of the planes and verify the original line. The conversation ends with the understanding that the equations of the planes can be derived from the parametric equations of the line.
Ocata
Hello,Suppose I have a vector equation:

\begin{cases}
x=0+10t\\
y=0+10t\\
z=0+10t
\end{cases}

Which forms the symmetric equation $\frac{x-0}{10}=\frac{y-0}{10}=\frac{z-0}{10}$

Now, I know the symmetric equations can be split up so that you can form the two planes whose intersection yields the initial vector:

$\frac{x-0}{10}=\frac{y-0}{10}$ and $\frac{y-0}{10}=\frac{z-0}{10}$

but I haven't been able to find any examples on how to get from the split symmetric equations of the line to two separate equations of a plane in standard form.

Would I just cross multiply each?

and get

Plane 1: x = y or x - y + 0z = 0
Plane 2: y = z or 0x + y - z = 0

Last edited:
Yes, that is what you want. More generally, given the line with parametric equations x= At+ a, y= Bt+ b, z= Ct+ d, we can write
$$t= \frac{x- a}{A}= \frac{y- b}{B}= \frac{z- c}{C}$$

Break that into the two equations
$$\frac{x- a}{A}= \frac{y- b}{B}$$
and
$$\frac{y- b}{B}= \frac{z- c}{C}$$

Multiply the first equation by AB to get B(x- a)= A(y- b) which is the same as Bx- Ay= aB- bA.
Multiply the second equation by BC to get C(y- b)= B(z- c) which is the same as Cy- Bz= bC- cB.

Those are the equations of the two planes that intersect in the original line.

Thank you. Now I'm going to work in reverse on my own and see if the intersection of the two planes via cross product yields the original line.

Appreciate your clarification on this topic.

## 1. How are equations of planes related to symmetric equations of a line?

Equations of planes and symmetric equations of a line are related because both represent mathematical representations of lines and planes in three-dimensional space. Equations of planes provide a general equation for a plane, while symmetric equations of a line provide a more specific representation of a line in space.

## 2. What is the difference between equations of planes and symmetric equations of a line?

The main difference between equations of planes and symmetric equations of a line is their level of specificity. Equations of planes provide a general equation for any plane in space, while symmetric equations of a line represent a specific line in space with a specific direction and point of origin.

## 3. How do you convert a symmetric equation of a line into an equation of a plane?

To convert a symmetric equation of a line into an equation of a plane, you can use the direction vector of the line and a point on the line to determine the normal vector of the plane. Then, you can plug in the normal vector and any point on the line into the general equation of a plane (Ax + By + Cz + D = 0) to get the specific equation of the plane.

## 4. Can equations of planes and symmetric equations of a line be used to find the intersection of a line and a plane?

Yes, equations of planes and symmetric equations of a line can be used to find the intersection point between a line and a plane. By setting the equations equal to each other and solving for the variables, you can find the coordinates of the point where the line and plane intersect.

## 5. What are some real-world applications of equations of planes and symmetric equations of a line?

Equations of planes and symmetric equations of a line have many real-world applications, including in engineering, architecture, and computer graphics. They can be used to calculate the slope and direction of a line, determine the orientation of a plane, and even model the flight paths of airplanes. They are also commonly used in 3D modeling and animation software to create realistic representations of objects and landscapes.

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