# Parametric and symmetric equations

• tony873004
In summary, the conversation discusses finding the parametric and symmetric equations of the line of intersection between the planes x+y+z=1 and x+z=0. The normal vectors of the planes and their cross product are used to determine the equation of the line. Setting z=0 yields the point of intersection and the parametric equation is found to be x=t, y=1, z=-t. The symmetric equations are also determined, but care must be taken to avoid dividing by zero. The conversation also includes a clarification and correction of a previous mistake.
tony873004
Gold Member
Find the parametric and symmetric equations of the line of intersection of the planes x+y+z=1 and x+z=0.

I got the normal vectors, <1,1,1> and <1,0,1> and their cross product <1,0,-1> or i-k.

I set z to 0 and got x=0, y=1, z=0.

How do I form parametric equation out of this?? I know it's x=t, y=1, z=-t because this problem is nearly identical to one from lecture. But how did he do that step?

This would make the symmetric equations x/1=y-1/0=z/-1. But I can't divide by 0, can I?

You set z = 0.

$$\pi_1: x+y=1$$
$$\pi_2: x=0$$

$$P(0,1,0)$$

The equation of the line is given by: $$\overrightarrow{r}=\overrightarrow{r}_0+t\overrightarrow{v}$$

And we know that the cross product of the two normal vectors of the plane is parallel to the line of intersection.

$$\overrightarrow{r}=<0,1,0>+t<1,0,-1>$$

$$x=-z;y=y_0$$

Since we don't write 0 under the denominator.

If $$x=x_0, y=y_0$$ vertical plane and $$z=z_0$$ horizontal plane.

Last edited:
Thanks for the explanation.

I don't get this:
$$\pi_2: x=1$$
If x+z=0 and I set z=0, then x+0=0. x=0 and 0+y+0=1, so y=1, hence P(0,1,0)

tony873004 said:
Thanks for the explanation.

I don't get this:
$$\pi_2: x=1$$
If x+z=0 and I set z=0, then x+0=0. x=0 and 0+y+0=1, so y=1, hence P(0,1,0)
Oh my. Sorry, I'm blind! I fixed it though.

Thanks. You explanation makes sense now.

## 1. What is the difference between parametric and symmetric equations?

Parametric equations describe a set of coordinates in terms of one or more independent variables, while symmetric equations describe a shape or curve in terms of a single equation with no independent variables.

## 2. How do you convert a parametric equation to a symmetric equation?

To convert a parametric equation to a symmetric equation, eliminate all independent variables by solving for one variable in terms of the other(s) and then substituting the resulting expression into the original parametric equation.

## 3. What are the advantages of using parametric equations?

Parametric equations allow for more flexibility and precision in describing complex shapes and curves, as well as making it easier to analyze and manipulate their properties. They also make it possible to create animations and visualizations of mathematical concepts.

## 4. Can parametric and symmetric equations be used interchangeably?

No, parametric and symmetric equations have different purposes and uses. While some shapes can be described by both types of equations, they are not equivalent and cannot always be used interchangeably.

## 5. What are some real-world applications of parametric and symmetric equations?

Parametric and symmetric equations are used in fields such as engineering, physics, and computer graphics to model and analyze various phenomena, such as motion, rotations, and geometric shapes. They are also used in creating visual effects in movies and video games.

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