Are the following relations correct? (vec a×vec b)·vec…

Are the following relations correct? $(\vec a\times\vec b)\cdot\vec c=-(\vec a\times\vec c)\cdot\vec b=-(\vec c\times\vec b)\cdot\vec a$.

Step 1: Rules

With $[\vec u,\vec v,\vec w]=(\vec u\times\vec v)\cdot\vec w$:

• a cyclic shift does not change the value: $[\vec a,\vec b,\vec c]=[\vec b,\vec c,\vec a]=[\vec c,\vec a,\vec b]$;
• swapping two vectors changes the sign.

Step 2: Checking the chain

\[ -(\vec a\times\vec c)\cdot\vec b=-[\vec a,\vec c,\vec b]=[\vec a,\vec b,\vec c]\quad(\text{one swap } \vec b\leftrightarrow\vec c), \]

\[ -(\vec c\times\vec b)\cdot\vec a=-[\vec c,\vec b,\vec a]=[\vec a,\vec b,\vec c]\quad(\text{one swap } \vec a\leftrightarrow\vec c). \]

\[\text{Both relations are correct.}\]

Numerical check with $\vec a=\vec i$, $\vec b=\vec j$, $\vec c=\vec k$: $1=-(-\vec j)\cdot\vec j=-(-\vec i)\cdot\vec i=1$ $\checkmark$