First question
I am given the question:
The plane ABC has equation: $$r\cdot(a \times b+b \times c+c \times a).$$
I know that $a\cdot (b×a) =a \cdot (a \times c)=0$ since you are doing a dot product with a vectors that are perpendicular to each other, but doesn't $b \times c$ also give a perpendicular vector to $a$?
Second question:
How can I write: $(b-a) \times (c-a)$ as $n = a \times b + b \times c + c \times a$? The answer I'm given sort of expands it like so:
$$
\begin{align}
n & = (b-a) \times (c-a) \\
& = (b-a) \times c-a) \\
& = b \times c - a \times c - b \times a + a \times a \\
& = a \times b + b \times c+ c \times a.
\end{align}
$$
But I don't know how it is expanded like that.