Answer
$c) 3$
$d)-3$
$e)-3$
$f) 0$
$a)-3$
b) 3
Work Step by Step
It's given that $3=(\vec{w} \times \vec{v})\cdot \vec{u} $
a) Since $-\vec{w} \times \vec{v}=\vec{v} \times \vec{w}$, we get that:
\[
\begin{aligned}
-3=\vec{u} \cdot(\vec{w} \times \vec{v}) &=\vec{u} \cdot(-\vec{v} \times \vec{w}) \\
&=-\vec{u} \cdot(\vec{v} \times \vec{w})
\end{aligned}
\]
b) Since $\vec{b} \cdot \vec{a}=\vec{a} \cdot \vec{b}$, we get that
\[
3=\vec{u} \cdot(\vec{v} \times \vec{w})=(\vec{v} \times \vec{w}) \cdot \vec{u}
\]
From the property
\[
\vec{u} \cdot(\vec{w} \times \vec{v})=\vec{v} \cdot(\vec{u} \times \vec{w})=\vec{w} \cdot(\vec{v} \times \vec{u})
\]
We get that:
c) $3=\vec{w} \cdot(\vec{u} \times \vec{v})=\vec{u} \cdot(\vec{v} \times \vec{w})$
d) $-3=-\vec{v} \cdot(\vec{u} \times \vec{w})=-\vec{u} \cdot(\vec{w} \times \vec{v})=\vec{v} \cdot(\vec{w} \times \vec{u})$
e) $-3=(\vec{u} \times \vec{w}) \cdot \vec{v}=\vec{v} \cdot(\vec{u} \times \vec{w})=-\vec{u} \cdot(\vec{v} \times \vec{w})$
f) $(\vec{w} \times \vec{w})\cdot \vec{v} =0$ because $\vec{w} \times \vec{w}=0$ for all vectors.
