# How to 'expand' cross product, and why is $a \cdot b \times a =a \cdot a \times c=0$ but $a \cdot b \times c$ is not?

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.

• First, note that $(b\times a)$ is a vector which is perpendicular to both $b$ and $a$. In particular it is perpendicular to $a$. That means by definition of perpendicular that $a\cdot (b\times a)=0$. Similarly for $a\cdot (a\times c)$. Remember the definition of perpendicular is that $u$ and $v$ are perpendicular if and only if $u\cdot v = 0$ Jun 20, 2018 at 19:26
• As far as why $a\cdot (b\times c)$ is not zero, it very well could be zero depending on the situation, however in the general case $b\times c$ is a vector perpendicular to both $b$ and $c$. We know nothing about whether or not $b\times c$ is perpendicular or not to $a$. Jun 20, 2018 at 19:28
• @JMoravitz What if $a$,$b$, and $c$ are vectors that all lie on the same plane in 3D space? would $a⋅(b×c)$ be zero then? Jun 20, 2018 at 19:30
• Assuming $b$ and $c$ are not multiples of one another in which case $b\times c$ is simply the zero vector in which case the result follows trivially, yes in that case $a\cdot (b\times c)$ would indeed be zero since $b\times c$ would give the normal vector to the plane which would be perpendicular to every vector in the plane including $a$. Jun 20, 2018 at 19:32
• @JMoravitz I might qualify that a bit. $a\cdot(b\times c)$ will be zero when the three points are on the same plane through the origin.
– amd
Jun 20, 2018 at 19:59