Finding Angle Between Planes

Question

I'm looking at an example to find the angle between two planes, and I figured you just transfer them into vectors normal to the plane and use the dot product.

I did this $\dfrac{\vec{A}\cdot \vec{B}}{|\vec{A}||\vec{B}|}$.

The teacher took the absolute value of the dot product, why?

$v_1 = \langle\sqrt{(5)}, 4, -2\rangle.$

$v_2 = \langle -\sqrt{(5)}, 2, 2\rangle.$

The answer is not $\cos ? = \dfrac{-1}{5\sqrt{(13)}}$ it is positive.

Why not?

Answer 1

You might define your normal vectors in such way that their dot product will be negative. It means the angle between planes will be obtuse. However, you probably needed an acute angle, so your teacher has applied the equality:

$$\cos(\theta) = -cos(\pi - \theta)$$