I am presented with the following problem:
I have a parallelepiped with adjacent edges $\vec{u} = [3,2,1]\\ \vec{v} = [2,3,1]\\ \vec{w} = [1,3,3]$
a) Find volume
b) find area of face determined by $\vec{u}$ and $\vec{w}$
c) find angle between $\vec{u}$ and face determined by $\vec{v}$ and $\vec{w}$.
So for (a), I just used a simple equation:
$V = ||\vec{u} \cdot (\vec{v} \times \vec{w}) ||,$ which gave me $V=11 \space \text{cubic units}$.
For (b), i found what $\vec{u} \times \vec{w}$ was, which is $[-7,-8,7]$. But from here, I'm not exactly sure how to find the area?
For part (c), what I did was find $\vec{v} \times \vec{w}$, then did $\vec{u} \times [ \vec{v} \times \vec{w} ]$.. which gave me a vector $[17, -3, -21]$. Then I solved for the angle by rearranging the equation
$||\vec{u} \times \vec{w}|| = ||\vec{u}||||\vec{w}||sin(\theta)$
Is that correct?