If A→,B→,C→ are vectors such that |B→|=|C→|. Then ((A→+B→)×(A→+C→))×(B→×C→)·(B→+C→)=.

((A→+B→)×(A→+C→))×(B→×C→)·(B→+C→)=[(A→+B→)·(B→×C→)(A→+C→)-(A→+C→)·(B→×C→)(A→+B→)]·(B→+C→)=[A→B→C→](C→-B→)·(B→+C→)=[A→B→C→]|C→|2-|B→|2=0

If A→,B→,C→ are vectors such that |B→|=|C→|. Then ((A→+B→)×(A→+C→))×(B→×C→)·(B→+C→)=.

((A→+B→)×(A→+C→))×(B→×C→)·(B→+C→)=[(A→+B→)·(B→×C→)(A→+C→)-(A→+C→)·(B→×C→)(A→+B→)]·(B→+C→)=[A→B→C→](C→-B→)·(B→+C→)=[A→B→C→]|C→|2-|B→|2=0

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If $\vec{A}, \vec{B}, \vec{C}$ are vectors such that $| \vec{B} | = | \vec{C} | .$ Then $((\vec{A} + \vec{B}) \times (\vec{A} + \vec{C})) \times (\vec{B} \times \vec{C}) \cdot (\vec{B} + \vec{C})$ =.

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$((\vec{A} + \vec{B}) \times (\vec{A} + \vec{C})) \times (\vec{B} \times \vec{C}) \cdot (\vec{B} + \vec{C})$

$= [(\vec{A} + \vec{B}) \cdot (\vec{B} \times \vec{C}) (\vec{A} + \vec{C}) - (\vec{A} + \vec{C}) \cdot (\vec{B} \times \vec{C}) (\vec{A} + \vec{B})] \cdot (\vec{B} + \vec{C})$

$= [\vec{A} \vec{B} \vec{C}] (\vec{C} - \vec{B}) \cdot (\vec{B} + \vec{C}) = [\vec{A} \vec{B} \vec{C}] [| \vec{C} |^{2} - | \vec{B} |^{2}] = 0$

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If A→,B→,C→ are vectors such that |B→|=|C→|. Then ((A→+B→)×(A→+C→))×(B→×C→)·(B→+C→)=.

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If $\vec{A}, \vec{B}, \vec{C}$ are vectors such that $| \vec{B} | = | \vec{C} | .$ Then $((\vec{A} + \vec{B}) \times (\vec{A} + \vec{C})) \times (\vec{B} \times \vec{C}) \cdot (\vec{B} + \vec{C})$ =.