If \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) are vectors such that |\(\vec{B}\)| = |\(\vec{C}\)|, then [(\(\vec{A}\)+\(\vec{B}\))x(\(\vec{A}\)+\(\vec{C}\))] x (\(\vec{B}\)x\(\vec{C}\)).(\(\vec{B}\)+\(\vec{C}\)) is equal to : 

0

We have (\(\vec{A}\)+\(\vec{B}\))x(\(\vec{A}\)+\(\vec{C}\)) = \(\vec{A}\)x\(\vec{C}\)+\(\vec{B}\)x\(\vec{A}\)+\(\vec{B}\)x\(\vec{C}\)

[(\(\vec{A}\)+\(\vec{B}\))x(\(\vec{A}\)+\(\vec{C}\))]x(\(\vec{B}\)x\(\vec{C}\))=[\(\vec{A}\).(\(\vec{B}\)x\(\vec{C}\))](\(\vec{C}\)-\(\vec{B}\))

[(\(\vec{A}\)+\(\vec{B}\))x(\(\vec{A}\)+\(\vec{C}\))]x(\(\vec{B}\)x\(\vec{C}\))(\(\vec{B}\)+\(\vec{C}\))=[\(\vec{A}\).(\(\vec{B}\)x\(\vec{C}\))](\(\vec{C}\)-\(\vec{B}\))(\(\vec{B}\)+\(\vec{C}\))

\(\vec{A}\)(\(\vec{B}\)x\(\vec{C}\)) [|\(\vec{C}\)|²-|\(\vec{B}\)|²] = 0          (Given : |\(\vec{B}\)| = |\(\vec{C}\)|)