Angle between planes has mysterious absolute value

**mathDad** · November 20th 2012, 05:41 PM

The angle between vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as follows.

$\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\left|\left| \mathbf{u} \right|\right| \left|\left| \mathbf{v} \right|\right|}$

It's derivation is straightforward.

The angle between two planes is equal to the angle between the planes' normal vectors, $\mathbf{n}_1$ and $\mathbf{n}_2$ . But then the book says the angle between the two planes is

$\cos \theta = \frac{\left| \mathbf{n}_1 \cdot \mathbf{n}_2 \right|}{\left|\left| \mathbf{n}_1 \right|\right| \left|\left| \mathbf{n}_2 \right|\right|}$

I assume the absolute value in the numerator is a because we want to take the acute angle between the two lines. Is that right? If not, why is it there? How does the absolute value represent the acute angle?

**Plato** · November 20th 2012, 05:56 PM

Originally Posted by mathDad

The angle between vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as follows.
$\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\left|\left| \mathbf{u} \right|\right| \left|\left| \mathbf{v} \right|\right|}$
It's derivation is straightforward.

The angle between two planes is equal to the angle between the planes' normal vectors, $\mathbf{n}_1$ and $\mathbf{n}_2$ . But then the book says the angle between the two planes is
$\cos \theta = \frac{\left| \mathbf{n}_1 \cdot \mathbf{n}_2 \right|}{\left|\left| \mathbf{n}_1 \right|\right| \left|\left| \mathbf{n}_2 \right|\right|}$

I assume the absolute value in the numerator is a because we want to take the acute angle between the two lines. Is that right? If not, why is it there? How does the absolute value represent the acute angle?

That is correct.
The angles between the two normals made accute,

**mathDad** · November 20th 2012, 06:24 PM

Originally Posted by Plato

That is correct.
The angles between the two normals made accute,

But why does the absolute value represent the acute angle? How do they get that?

**Plato** · November 20th 2012, 06:35 PM

Originally Posted by mathDad

But why does the absolute value represent the acute angle? How do they get that?

Actually I have a quibble with your notation.

I think it should be $\theta=\arccos\left(\frac{|n_1\cdot n_2|}{\|n_1\|\|n_2\|}\right)$ .
Now the $\arccos$ function returns an acute angle values for positive arguments.

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