# Angle Between Two Planes

• Bassa
In summary: The acute angle is the one that is used in the formula for the angle between two planes. So even if the planes intersect at an angle greater than 90°, we can always find the smaller acute angle between them.

## 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?

Bassa said:

## 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
LCKurtz said:
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!

LCKurtz said:
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.

Bassa said:
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°.