volume of the parallepiped spanned by the vectors

Question

Hi I am having difficulty with part (2) of the following proposition.

Suppose that $x,y,z\in\mathbb{R}^3$, then

(1) $\|x\times y\|=\|x\|\|y\|\sin\theta$ is the area of the parallelogram spanned by the vectors $x$ and $y$, where $\theta$ is the angle between them.

(2) $$ |(x\times y)\cdot z|=\left|\text{det}\left( \begin{array}{ccc} z_1 & z_2 & z_3 \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{array} \right)\right|$$ is the volume of the parallelepiped spanned by th vectors.

The solution is the following

I am fine with everything except the last two lines of the proof of (2). Could anyone help me to see how could $$A\cdot\frac{(x\times y)\cdot z}{\|x\times y\|}=A\|z\|\cos \alpha?$$

Answer 1

$u \cdot v = \|u\|\|v\|\cos \alpha$, where $\alpha$ is the angle between $u$ and $v$ is a well-known formula. Indeed, I have often seen it used as the definition of the dot product. In this case $ u = x \times y$ and $v = z$.