Basic vector Calculation

Question

let $ (\vec a\times\vec b)\cdot {\vec c} = 1$

$ \vec u = \vec a + \vec b$

$\vec v = \vec b + \vec c$

$\vec w = \vec c + \vec a$

Calculate $(\vec u\times\vec v)\cdot{\vec w} = ?$

I dont know how to solve this question how to use the information that $ (\vec a\times\vec b)\cdot{\vec c} = 1$

Answer 1

We have that

$$\vec u \times \vec v =\vec a \times \vec b+\vec a \times \vec c+\vec b \times \vec c $$

and

$$(\vec u \times \vec v)\cdot \vec w =(\vec u \times \vec v)\cdot \vec a+(\vec u \times \vec v)\cdot \vec c=(\vec b \times \vec c)\cdot \vec a+(\vec a \times \vec b)\cdot \vec c$$

then recall that by scalar triple product

$$(\vec b \times \vec c)\cdot \vec a=(\vec a \times \vec b)\cdot \vec c$$