A long plank, with a $1 \times 1$ cross section, is cut as shown below. The region of the cut is a parallelogram with sides $\sqrt{2}$ and $\sqrt{3}.$ Find the area of the parallelogram.
I recognize that I could calculate the area of the parallelogram using vector cross products, but I don't know how to proceed. Any help would be much appreciated!
On
lets put the start of the cut at the origin.
Before it is cut, the plank extends indefinitely in the z-direction and 1 inch in the positive x and 1 inch in the positive y-direction. That is, the sides of the plank are the xz and yz planes.
vector $u, v = (1,0,z_1), (0,1, z_2)$ represent the sides of the paralellogram.
$v$ has length $\sqrt 3$
$0^2 + 1^2 + z_2^2 = 3\\ z_2 = \sqrt2$
$u$ has length $\sqrt 2.$
$z_1 = 1$
Now that you have $u,v$ you can find $u\times v$
Hint:
The idea with the vector product is the correct one.
The vectors are $(1,0,1)$ and $(0,1,2)$.
Can you take it from here?