# Find the line of intersection between 2 Planes

• prace
In summary, to find the line of intersection between two planes, you can either use the cross product of the normal vectors or solve for two variables in terms of the third and use that as a parameter for the parametric equations of the line.
prace
Hello,

I am trying to find the line of intersection between these two planes:

$$P_1$$ = x + 2y -9z = 7
$$P_2$$ = 2x - 3y + 17z = 0

I found the direction vector needed for the line of intersection between these two points by taking the cross product of the $$P_1$$ normal vector and the $$P_2$$ normal vector which gave me $$\vec{a}$$ = <7,-35,-7>

Now all I need is a point somewhere along the direction of that vector. This is where I am stuck. Any help on finding this point would be awesome. Thanks!

~Peter

Why find the equations of the line that way? Since you have two equations for three unknown variables, just solve for two of them in terms of the third, then use that third variable as parameter.

For example, to find equations of the line of intersection of x+ y+ z= 1 and 2x- y+ z= 0, adding the equations gives 3x+ 2z= 1 so z= 1/2- (3/2)x.
Then y= 1- x- z= 1- x- 1/2+ (3/2)z= 1/2+ (1/2)x. Let x= t. Then the parametric equations are x= t, y= 1/2+ (1/2)t, z= 1/2- (3/2)t. In vector form, $$\vec{r}= t\vec{i}+ (1/2+ (1/2)t)\vec{j}+ (1/2- (3/2)t)\vec{k}$$ or $$\vec{r}= ((1/2)\vec{j}+ (1/2)\vec{k})+ (\vec{i}+ (1/2)\vec{j}+ (1/2)\vec{k})t$$.

Hello Peter,

Finding the line of intersection between two planes involves finding the point where the two planes intersect. To do this, you can set the equations of the planes equal to each other and solve for two variables. In this case, you can set x + 2y - 9z = 7 equal to 2x - 3y + 17z = 0 and solve for x and y. This will give you a point on the line of intersection, which you can then use to find other points on the line by adding multiples of the direction vector you found. I hope this helps. Let me know if you have any other questions.

## What is the definition of the line of intersection between 2 planes?

The line of intersection between 2 planes is the line where the two planes intersect each other. It is the set of points that are common to both planes.

## How do you find the line of intersection between 2 planes?

To find the line of intersection between 2 planes, you can use the method of substitution or elimination. This involves finding the values of the variables that satisfy both plane equations, which will give you the coordinates of the points on the line of intersection.

## What information do you need to find the line of intersection between 2 planes?

You will need the equations of both planes in order to find the line of intersection. These equations should be in standard form, where the variables are on one side and the constant is on the other side.

## Can the line of intersection between 2 planes be parallel?

Yes, the line of intersection between 2 planes can be parallel if the planes are parallel to each other. In this case, the line of intersection will be a degenerate line, meaning it is a line that only consists of one point.

## What does the slope of the line of intersection between 2 planes represent?

The slope of the line of intersection between 2 planes represents the rate of change of the line in the x and y directions. It can also be interpreted as the direction in which the line is moving.

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