Question

Find the value of $\vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times(\vec{c}+\vec{a})+\vec{c}\times(\vec{a}+\vec{b})$.

Hint:

Whenever you see a sum of cross products like this, pair them using the property $\vec{p}\times\vec{q} = -\vec{q}\times\vec{p}$ to simplify quickly.

Answer 1

Step 1: Expand each term. \[ \vec{a}\times(\vec{b}+\vec{c}) = \vec{a}\times\vec{b} + \vec{a}\times\vec{c} \] \[ \vec{b}\times(\vec{c}+\vec{a}) = \vec{b}\times\vec{c} + \vec{b}\times\vec{a} \] \[ \vec{c}\times(\vec{a}+\vec{b}) = \vec{c}\times\vec{a} + \vec{c}\times\vec{b} \]

Step 2: Add them. \[ \vec{a}\times(\vec{b}+\vec{c}) + \vec{b}\times(\vec{c}+\vec{a}) + \vec{c}\times(\vec{a}+\vec{b}) \] \[ = (\vec{a}\times\vec{b} + \vec{a}\times\vec{c}) + (\vec{b}\times\vec{c} + \vec{b}\times\vec{a}) + (\vec{c}\times\vec{a} + \vec{c}\times\vec{b}) \]

Step 3: Use the property $\vec{p\times\vec{q} = -(\vec{q}\times\vec{p})$.} \[ \vec{a}\times\vec{b} + \vec{b}\times\vec{a} = 0, \vec{a}\times\vec{c} + \vec{c}\times\vec{a} = 0, \vec{b}\times\vec{c} + \vec{c}\times\vec{b} = 0 \]

Step 4: Final result. \[ \boxed{0} \]