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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?

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For b), the formula is $\|u \times w\|$. For c), find the angle between $u$ and $v \times w$ or $-v \times w$, whichever is acute, and subtract it from $90^\circ$ (draw a picture).

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    One more comment. No, draw the plane spanned by $v$ and $w$ head-on so it looks like a line. Sketch $v \times w$ perpendicular to the plane. Sketch $u$ in the case where the angle between $u$ and $v \times w$ is acute and in the case where it is obtuse. Figure out the angle between $u$ and the plane in each case.2012-06-10