Why is a+b+c = a-a+c?

Question

Why does $a+b+c = a-a+c$? I don't understand. Is it some math property that i didn't know of?

Answer 1

Note that we have $a+b=0$, so $b=-a$. Using that, we have $a+b+c=a-a+c$.

In general, $a+b+c=a-a+c$ is not true. It is true only if we know $b=-a$.

Answer 2

From second inner product : $$\begin {pmatrix} a & b & c \end{pmatrix} . \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = a.1+b.1+c.0$$

As the above two vectors are orthogonal then $a+b=0$

So substitute $b=-a$ in the equation which you mentioned.