Mixed vector product among three vectors

The main topic is the modification of variables in triple integrals

$$int int int_D f (x, y, z) ; dx ; dy ; dz = int int int_ {D & # 39;} f[x(u,v),y(u,v),z(u,v)] cdot || J || cdot ; you ; dv ; dw$$

If $$D & # 39;$$ is a normal domain like

$$a leq u leq b quad, quad alpha (u) leq v leq beta (u) quad, quad c leq w leq d$$

with change of variables

$$u = U quad, quad v = alpha (U) + V[beta(U) – alpha(U)] quad, quad v = omega$$

we have

$$int int int_D f (x, y, z) ; dx ; dy ; dz = int_a ^ b dU int_c ^ dd omega int_0 ^ 1 f (U, V, omega) cdot || J || cdot dV$$

Now my teacher slide says:

I can not understand this step. Ok, is a mixed vector product of three vectors: but how is it done?

thank you in advance

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