# Finding the Line that Represents the Intersection of Two Planes

• Jun 13th 2013, 09:38 AM
Bashyboy
Finding the Line that Represents the Intersection of Two Planes
Suppose we have two intersecting planes, P1 and P2, whose normal vectors are n1 and n2, respectively. If one was asked to find the line of intersection, one could simply calculate $\displaystyle \vec{n}_1 \times \vec{n}_2 = \vec{u}$, of which $\displaystyle \vec{u}$ would be the direction vector for the line.

My question is, why is it the case that calculating the cross product of the normal vectors of the planes provides the direction vector?
• Jun 13th 2013, 09:47 AM
Plato
Re: Finding the Line that Represents the Intersection of Two Planes
Quote:

Originally Posted by Bashyboy
Suppose we have two intersecting planes, P1 and P2, whose normal vectors are n1 and n2, respectively. If one was asked to find the line of intersection, one could simply calculate $\displaystyle \vec{n}_1 \times \vec{n}_2 = \vec{u}$, of which $\displaystyle \vec{u}$ would be the direction vector for the line.

My question is, why is it the case that calculating the cross product of the normal vectors of the planes provides the direction vector?

Because the line of intersection$\displaystyle ,~\ell,$ is perpendicular to each of the normals.
The vector $\displaystyle \vec{n}_1 \times \vec{n}_2$ is perpendicular to each of $\displaystyle \vec{n}_1 ~\&~ \vec{n}_2$.

Thus $\displaystyle \vec{n}_1 \times \vec{n}_2 = \vec{u}$ is the direction vector for $\displaystyle \ell$.
• Jun 13th 2013, 10:21 AM
Bashyboy
Re: Finding the Line that Represents the Intersection of Two Planes
So, the idea follows simply from a visual understanding?
• Jun 13th 2013, 10:30 AM
Plato
Re: Finding the Line that Represents the Intersection of Two Planes
Quote:

Originally Posted by Bashyboy
So, the idea follows simply from a visual understanding?

I don't know how to respond to that statement.

Any line in a plane is perpendicular to the plane's normal.

Because $\displaystyle \ell$ is the line of intersection, it is in both planes, it is perpendicular to each of $\displaystyle \vec{n}_1 ~\&~ \vec{n}_2$.

And that is $\displaystyle \vec{n}_1\times \vec{n}_2$.
• Jun 13th 2013, 11:01 AM
Bashyboy
Re: Finding the Line that Represents the Intersection of Two Planes
Oh, wow, I didn't think about it in that sense, that the line of intersection lies in the plane. Obviously that makes sense: for if it wasn't, then the coordinates on the line wouldn't satisfy the equation of the plane, and vice-versa.