How did it get from line 1 to line 2 to line 3 (Vector Cross Product)

Question

How does it get from line 1 to line 2 can someone explain it in verybbasic terms Thanks

Answer 1

The second $=$ expands the cross-product; like ordinary multiplication, it's linear in both arguments. The last $=$ uses antisymmetry $u\times v=-v\times u$, including the "diagonal" special case $u=v=a$, giving$$a\times a=-a\times a\implies 2a\times a=0\implies a\times a=0.$$

Answer 2

Remember that for any vectors $u,v$, we have $u \times v = -v \times u$ and $v \times v = 0$. So, $$ b \times c - a \times c - b \times a + a \times a =\\ b \times c - -(c \times a) --(a \times b) + 0 =\\ b \times c + c \times a + a \times b. $$