How Is the Angle Between Two Planes Calculated Using Vector Dot Product?

[Bassa]

Homework Statement

I am preparing for my calculus III class over the break. I came across the formula for the angle between two planes which is:

cosΘ = (|a.b|)/(||a|||b||)

Homework Equations

cosΘ = (|a.b|)/(||a|||b||)
a.b = ||a||||b||cosΘ

The Attempt at a Solution

I know that the dot product formula is:

a.b = ||a||||b||cosΘ

Why do we put the absolute value bars when we are trying to find the angle between two planes if the original formula doesn't have absolute values in it?

 

[LCKurtz]

Homework Statement

I am preparing for my calculus III class over the break. I came across the formula for the angle between two planes which is:

cosΘ = (|a.b|)/(||a|||b||)

Homework Equations

cosΘ = (|a.b|)/(||a|||b||)
a.b = ||a||||b||cosΘ

The Attempt at a Solution

I know that the dot product formula is:

a.b = ||a||||b||cosΘ

Why do we put the absolute value bars when we are trying to find the angle between two planes if the original formula doesn't have absolute values in it?

That is the formula for the angle between two vectors. You haven't told us what a and b are but I would presume you mean for them to be the normal vectors to the plane. Remember that the angle between two vectors can be from $0$ to $\pi$, but the angle between two planes in never greater than $\pi / 2$. If your cosine comes out negative that means the angle between the normals is greater than $\pi /2$ and you want its supplement. Dropping the negative sign will give you that.

 

[Bassa]

That is the formula for the angle between two vectors. You haven't told us what a and b are but I would presume you mean for them to be the normal vectors to the plane. Remember that the angle between two vectors can be from $0$ to $\pi$, but the angle between two planes in never greater than $\pi / 2$. If your cosine comes out negative that means the angle between the normals is greater than $\pi /2$ and you want its supplement. Dropping the negative sign will give you that.

Thank you very much for this thorough explanation!

 

[Bassa]

That is the formula for the angle between two vectors. You haven't told us what a and b are but I would presume you mean for them to be the normal vectors to the plane. Remember that the angle between two vectors can be from $0$ to $\pi$, but the angle between two planes in never greater than $\pi / 2$. If your cosine comes out negative that means the angle between the normals is greater than $\pi /2$ and you want its supplement. Dropping the negative sign will give you that.

Well, now come to think about it more intently, why can the angle between two planes not be more than 90 degrees? I could imagine many intersecting planes having an angle more than 90 degrees between them.

 

[Mark44]

Well, now come to think about it more intently, why can the angle between two planes not be more than 90 degrees? I could imagine many intersecting planes having an angle more than 90 degrees between them.

Because there is also an acute angle that is formed, that is the supplement of the angle of more than 90°.