Finding a third plane that has a dihedral angle to two other planes.

[user8899]

Homework Statement

The acute angle between two planes is called the dihedral angle. Plane x−3y+2z=0 and plane 3x−2y−z+3=0 intersect in a line and form a dihedral angle θ . Find a third plane (in point-normal, i.e. component, form) through the point (-6/7,0,3/7) that has dihedral angle θ/2 with each of the original planes. Do the three planes intersect at a point or in a line? Explain all steps carefully.

Homework Equations

cosθ=|n₁ (dot product) n₂| /|n₁||n₂|,

The Attempt at a Solution

I found the dihedral angle of the first two planes to be 1/2, but then I'm not sure what to do after that.

 

[LCKurtz]

Homework Statement

The acute angle between two planes is called the dihedral angle. Plane x−3y+2z=0 and plane 3x−2y−z+3=0 intersect in a line and form a dihedral angle θ . Find a third plane (in point-normal, i.e. component, form) through the point (-6/7,0,3/7) that has dihedral angle θ/2 with each of the original planes. Do the three planes intersect at a point or in a line? Explain all steps carefully.

Homework Equations

cosθ=|n₁ (dot product) n₂| /|n₁||n₂|,

The Attempt at a Solution

I found the dihedral angle of the first two planes to be 1/2, but then I'm not sure what to do after that.

Don't you mean you found the cosine of the dihedral angle to be 1/2? Wouldn't your new plane bisect the dihedral angle? Can you get its normal from the given normals?

 

[user8899]

Don't you mean you found the cosine of the dihedral angle to be 1/2? Wouldn't your new plane bisect the dihedral angle? Can you get its normal from the given normals?

Yes, that is what I meant. Do we add the two normals to get the third normal? I am really confused.

 

[LCKurtz]

Yes, that is what I meant. Do we add the two normals to get the third normal? I am really confused.

Think about it and draw some pictures. Say you have two vectors with their tails at the same place. Will their sum bisect the angle between them or do you have to have some condition on the vectors to make it happen?

 

[user8899]

Think about it and draw some pictures. Say you have two vectors with their tails at the same place. Will their sum bisect the angle between them or do you have to have some condition on the vectors to make it happen?

No because the vectors have to be tip to tail to add up?

 

[LCKurtz]

No because the vectors have to be tip to tail to add up?

No they don't. Think about the parallelogram rule for addition of vectors. The sum is the diagonal of the parallelogram. What has to be true about the two original vectors for that sum vector to bisect the angle between them?