$$(\vec{a}+\vec{b})\cdot[(\vec{b}+\vec{c})\times(\vec{c}+\vec{a})]=2\vec{a}\cdot(\vec{b}\times\vec{c})$$
Is there an option to prove it using the properties of dot and cross product? or do I need to take a general vectors to show it? can I use vectors in $\mathbb{R}^2$