Assume that $v$ and $w$ are linearly independent vectors in $\Bbb R^3$. Determine which of the following sets of vectors is linearly independent.
a) $\left\{v\times w,0,v\right\}$
b) $\left\{v,v\times w,v\times \left(v\times w\right)\right\}$
c) $\left\{v,v+w,v+2w\right\}$
d) $\left\{v+w,v-w\right\}$
e) $\left\{v\times w,2v,\:w\times v\right\}$
f) $\left\{v\times w,\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},v,w\right\}$
I am not entirely sure how to approach this problem. My intuition is telling me that $a$ is dependent because of the zero vector, $c$ is dependent because one row is a multiple of a another row, and I thought $f$ was dependent because $v$ and $w$ are vectors in $\Bbb R^3$ whereas that set has $4$ vectors.
But I know that I am supposed to find whether or not there is a non-trivial linear combination of vectors that equals zero. But I am not sure how to do that in this case as no numbers are have been provided.
Any help would be highly appreciated!
HINT
Note that
a) is not since it contains $\vec 0$
b) $\vec v \times (\vec v \times \vec w)=\vec 0$
c) $\vec u_2-\vec u_1=\vec v+\vec w-\vec v=\vec w$ and $\vec u_3-\vec u_2=\vec w$
d) prove that $a\vec u_1+b\vec u_2=0 \iff a=b=0$
e) $\vec v \times \vec w=-\vec w \times \vec v$
f) #vectors > dimension