# Intersection Line for Two Planes

• kieth89
In summary, the conversation was about finding an equation of the line where the planes Q and R intersect. The process involves finding a common point between the two planes and using the cross product of their normal vectors to determine the direction vector of the line. The final equation for the line is given by \vec{r}(t) = <-1 + 4t, t, 1 - 5t>, but there may be different starting points resulting in different parametric representations. However, strictly speaking, neither answer is an equation as there is no equals sign.
kieth89

## Homework Statement

Find an equation of the line where the planes Q and R intersect.
$Q: -2x + 3y - z = 1; R: x + y + z = 0$

## Homework Equations

Equation of a Plane: $ax + by + cz = d,$where $\vec{n} = <a, b, c>$
Equation of a Line in $R^{3}$: $\vec{r}(t)=<x_{0}, y_{0}, z_{0}> + t<x,y,z>$

## The Attempt at a Solution

First I find a point common to both planes, this will be $P_{0}$.
Set $y = 0$ and add the plane equations:
$-2x + 0y - z - 1 = 0$
$1x + 0y + z - 0 = 0$
Resulting in: $-x - 1 = 0$ so $x = -1 , z = 1$ and $P_{0} = (-1, 0, 1)$.

Now I find the direction vector for our line. This will just be the cross product of the normal vectors from the two plane equations:
$<-2, 3, -1> X <1, 1, 1> = <4, 1, -5>$

Now I just plug the obtained info into the equation for a line:
$\vec{r}(t) = <-1, 0, 1> + t<4, 1, -5> -> \vec{r}(t) = <-1 + 4t, t, 1 - 5t>$

I felt pretty confident in this answer, but the answer key says it should be $<-\frac{1}{5} + 4t, \frac{1}{5} + t, -5t>$. I'm wondering if my answer is different due to using a different $P_{0}$, but I don't know...

The answers are just different parametrisations of the line.
If we label your ##t## as ##t## and theirs as ##t'## then the conversion is ##t'=t-\frac{1}{5}##.

This is hinted at by the question asking you to 'find an equation of...' rather than 'find the equation of...'

Strictly speaking though, neither answer is correct, as neither is an equation - where is the equals sign? But there is no single equation that denotes the line. Two are needed.

kieth89 said:
Now I just plug the obtained info into the equation for a line:
$\vec{r}(t) = <-1, 0, 1> + t<4, 1, -5> -> \vec{r}(t) = <-1 + 4t, t, 1 - 5t>$

I felt pretty confident in this answer, but the answer key says it should be $<-\frac{1}{5} + 4t, \frac{1}{5} + t, -5t>$. I'm wondering if my answer is different due to using a different $P_{0}$, but I don't know...
Your equation is correct and also the given solution is really a parametric representation of the line of intersection. The difference is the starting point only. The equation of a line is given by ##\vec r(t) = \vec a + \vec b t##

andrewkirk said:
Strictly speaking though, neither answer is correct, as neither is an equation - where is the equals sign? But there is no single equation that denotes the line. Two are needed.
The answer of the OP is an equation. It has two sides and a "=" in between.

Yay! So it is due to different starting points. Thanks for the help everyone.

ehild said:
The answer of the OP is an equation. It has two sides and a "=" in between.
Oh my goodness, so it does! I don't know how I missed that. Either my mind's been playing tricks on me or my vision is deteriorating even faster than I feared.

## 1. What is the Intersection Line for Two Planes?

The Intersection Line for Two Planes refers to the line that is formed when two planes intersect in three-dimensional space. This line is the common line that exists between the two planes and can be described by a set of coordinates.

## 2. How is the Intersection Line calculated?

The Intersection Line is calculated by finding the point where the two planes intersect. This can be done by solving a system of equations that represent the two planes. The solution to this system of equations will give the coordinates of the point of intersection, which in turn defines the Intersection Line.

## 3. How many solutions are possible for the Intersection Line of Two Planes?

The Intersection Line of Two Planes can have three possible scenarios: no solution, a single solution, or infinitely many solutions. If the two planes are parallel, there will be no intersection and therefore no solution. If the two planes are the same, there will be infinitely many solutions as they are essentially the same plane. If the two planes intersect at a single point, there will be a single solution for the Intersection Line.

## 4. What is the significance of the Intersection Line in geometry and physics?

The Intersection Line is significant in both geometry and physics because it helps to define the relationship between two planes in three-dimensional space. In geometry, it is used to find the point of intersection between two lines, while in physics, it is used to calculate the intersection of two planes representing different forces or objects.

## 5. Can the Intersection Line for Two Planes be in any direction?

Yes, the Intersection Line for Two Planes can be in any direction, as long as it exists between the two intersecting planes. It can be vertical, horizontal, or at any angle in between. The direction of the Intersection Line is determined by the direction of the two intersecting planes and the point of intersection between them.

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