Find a constant $c$ such that $\omega \wedge \nu \wedge \gamma = cdx\wedge dy \wedge dz$

Question

Let $\omega = dx +5dy - dz, \nu = 2dx-dy+dz, \gamma = -dx+dy +2dz$

Find a constant $c$ such that $\omega \wedge \nu \wedge \gamma = cdx\wedge dy \wedge dz$

I multiplied it and ended up at $(-11dx\wedge dy+4d\wedge dz-3dz\wedge dx)\wedge (-dx+dy+2dz)$ but I haven't been taught the definition of multiplying 1-forms with 2-forms yet.. I do not know how to proceed or how I should have done this.

Answer 1

$a\wedge a=0$, so $dx\wedge dy\wedge dx=0$ and so on. So we have $$\begin{aligned}&(-11dx\wedge dy+4dy\wedge dz-3dz\wedge dx)\wedge (-dx+dy+2dz)\\&=-22dx\wedge dy\wedge dz-4dy\wedge dz\wedge dx-3dz\wedge dx\wedge dy\\ &=-22dx\wedge dy\wedge dz+4dy\wedge dx\wedge dz+3dx\wedge dz\wedge dy\\ &=-22dx\wedge dy\wedge dz-4dx\wedge dy\wedge dz-3dx\wedge dy\wedge dz\\ &=-29dx\wedge dy\wedge dz \end{aligned} $$