I The angle of intersection between two planes in R3

1. Aug 17, 2016

Erenjaeger

What is the angle of intersection between the two planes in ℝ3 with general equations
x-y=5 and y-z=7?

I know that the angle between then is equal to cos-1 (u⋅v/||u|| ||v||) but I am stuck on the general equations given, how can I solve when given these ?

2. Aug 17, 2016

Staff: Mentor

Find a normal to each plane, and then use the dot product to find the cosine of the angle between the two normals. That will be the same as the angle between the two planes.

3. Aug 17, 2016

Erenjaeger

how would i find a normal to each plane?

4. Aug 17, 2016

Erenjaeger

im confused with those general equations given in the problem

5. Aug 17, 2016

Erenjaeger

In my course book, we are given an example where we have two points in ℝ3 which are (2,-1,3) and is parallel to (3,2,-2) so the parametric equations are
x=2+3t
y=-1+2t
z=3-2t
where t∈ℝ
solving for t gives

t=x-2/3, t=y+1/2, t=3-z/2

and this yields two equations
x-2/3 = y+1/2 and y+1/2 = 3-z/2 which I can understand because they're all the same value which is 't' so they are all equal to one another right?

the part where im lost is where it says rearranging these two equations gives some general equations for the two planes...
2x-3y=7 and y+z=2
how do they rearrange to get those equations ?

6. Aug 17, 2016

chiro

Hey Erenjaeger.

This is done by solving a linear system (i.e. a "matrix") and finding what occurs when you reduce it down to "row echelon form".

Have you ever done linear algebra before?

7. Aug 17, 2016

Erenjaeger

yeah

8. Aug 17, 2016

chiro

It's basically just an application of standard matrix methods.

9. Aug 17, 2016

Erenjaeger

okay, ill read through that section in my course book, thanks

10. Aug 17, 2016

Staff: Mentor

Given a plane in R3, Ax + By + Cz = D, a normal to this plane is the vector <A, B, C>. My suggestion, which doesn't involve matrices, is a lot simpler than the one chiro gave.

Find a normal to each plane: $\vec{n_1} = <A_1, B_1, C_1>$ and $\vec{n_2} = <A_2, B_2, C_2>$ and use the fact that $\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}|\cos(\theta)$, where $\theta$ is the angle between the two vectors.