How would one go about resolving the vector $\vec{p}$ into parallel and perpendicular vectors to the given vector $\vec{w}$
By considering - $\vec{w}\times(\vec{p}\times\vec{w})$
So far I have used the triple vector product however I seem to just get zero when I do this so I feel like i'm making a mistake somewhere.
Inasmuch as $\vec p\times \vec w$ is perpendicular to both $\vec p$ and $\vec w$, we can decompose $\vec p$ as
$$\begin{align} \vec p&=A\vec w+B[\vec w\times(\vec p\times \vec w)]\tag1 \end{align}$$
Note that $\vec w\times(\vec p\times \vec w)$ is perpendicular to $\vec w$.
Taking the inner product of $\vec p$ with $\vec w$, we find from $(1)$ that
$$A=\frac{\vec p\cdot \vec w}{|\vec w|^2} $$
Taking the vector product of $\vec p$ with $ \vec w$, we find from $(1)$ that
$$B=\frac{1}{|\vec w|^2}$$
Hence, denoting the unit vector along $\vec w$ as $\hat w=\frac{\vec w}{|\vec w|}$
$$\vec p=(\vec p\cdot \hat w)\hat w+ ( \hat w \times\vec p)\times \hat w$$