$P\cdot (Q \times P)$ where $P$ and $Q$ are vectors

Question

The answer is zero, but why?

my theory is that $P.Q$ is a scalar product, you cant do cross product between the remaining vector and scalar

but it was written in the answer that the cross product of vector would be parallel to the parallelogram of the vector and therefore parallel to $P$. and the dot product $P$ with another Parallel vector would be zero

so which one is the correct method?

(the question does not specify which is comes first- $(P\cdot Q)\times P$ or $P\cdot (Q \times P)$ in case it was relevant)

Answer 1

There are two ways we might associate terms, either as $(P \cdot Q) \times P$, or as $P \cdot (Q \times P)$. Fortunately, the first one doesn't make sense, we would be crossing a vector with a scalar (ugh, how many times have I heard that joke?). So the correct way to interpret it is to take it as $P \cdot (Q \times P)$, which correctly dots a vector with another vector.

To see why this quantity is zero, remember that the cross product of two vectors returns a vector that is orthogonal (perpendicular) to both of them. So $Q \times P$ is orthogonal to both $Q$ and $P$. Thus $P \cdot (Q \times P)$ is a dot product between two orthogonal vectors. Do you remember what the result is always when you dot two orthogonal vectors?

Answer 2

$Q\times P$ is perpendicular to the plane spanned by $P$ and $Q$, so $P\cdot(Q\times P)=0$.

You are right that $(P\cdot Q)\times P$ does not make sense since $P\cdot Q$ is a scalar.