Line of intersection between two perpendicular planes

[MarkSA]

Hello,

Q: Find the line of intersection between perpendicular planes:
5x-3y+z=4
x+4y+7z=1

Normal vectors: <5,-3,1> and <1,4,7> .. dot product is zero which is why they are perpendicular. The cross product of the normal vectors should be parallel to my line of intersection. I'm not entirely sure why. For the resultant vector from crossing, I get: <-25,-34,23>

So I need a position vector and direction vector to make my line.. right? For position, the points on the line must be present in both planes if it's intersecting the planes. But how do I get those points? I thought about setting x=0 in both equations and solving for y,z, but how can I know that the line actually crosses the x-axis at some point? Is it nothing to worry about unless the direction vector happens to be say: <0,-34,23>?

If I set x=0 I get: -3y+z=4 and 4y+7z=1. or y=-27/25 and z=19/25... so I would end up with:
x = -25t, y = -27/25 - 34t, z = 19/25 + 23t

Am I doing this correctly?

 

[galactus]

You have the idea.

\(\displaystyle n_{1}=[5,-3,1], \;\ n_{2}=[1,4,7]\) are normals to the planes.

\(\displaystyle n_{1}\times n_{2}=[-25,-34,23]\) is parallel to the line of intersection.

Let z=0 in each equation and solve for x and y and we get \(\displaystyle x=\frac{19}{23}, \;\ y=\frac{1}{23}\)

So, we have parametric equations:

\(\displaystyle x=\frac{19}{23}-25t, \;\ y=\frac{1}{23}-34t, \;\ z=23t\)

 

[pka]

The above solution is correct.
However, I need to add some general remarks.
To write the line of intersection of two intersecting planes here are the steps.
Find a point on both planes.
The cross product of the two normals is the direction vector of the line.
Write the equation using the point and vector from above.

 

[MarkSA]

Can you explain how/why z=0 was chosen rather than x=0 or y=0? I notice that the line equation differs depending which one is chosen to be set to zero in the problem.

 

[pka]

Can you explain how/why z=0 was chosen rather than x=0 or y=0? I notice that the line equation differs depending which one is chosen to be set to zero in the problem.

That cannot possibly be true! You need to check your arithmetic.
One does get different points, but the same line.