p→$$=b$$→×c→[a→b→c→],q→$$=c$$→×a→[a→b→c→] and r→$$=a$$→×b→[a→b→c→] where a→,b→ and c→ are thee $$non-coplanar$$ vectors, then the value of the expression (a$$→$$+b$$→$$+c$$→$$)⋅(p$$→$$+q$$→$$+r$$→$$) is

Question

p→$$=b$$→×c→[a→b→c→],q→$$=c$$→×a→[a→b→c→] and r→$$=a$$→×b→[a→b→c→] where a→,b→ and c→  are thee $$non-coplanar$$ vectors, then  the value of the expression (a$$→$$+b$$→$$+c$$→$$)⋅(p$$→$$+q$$→$$+r$$→$$) is

Infinity Learn · Accepted Answer

We have,a→⋅p→$$=a$$→⋅(b$$→×c→$$)[a→b→c→]$$=a$$→⋅(b$$→×c→$$)[a→b→c→]$$=$$[a→b→c→][a→b→c→]$$=1a$$→⋅q→$$=a$$→⋅c→×a→[a→b→c→]$$=$$[a→c→a→][a→b→c→]$$=0$$ Similarly, a→⋅r→$$=0$$,b→⋅p→$$=0$$,b→⋅q→$$=1$$,b→⋅r→$$=0c$$→⋅p→$$=0$$,c→⋅q→$$=0$$ and c→⋅r→$$=1$$∴ (a$$→$$+b$$→$$+c$$→$$)⋅$$(p$$→$$+q$$→$$+r$$→$$)=a$$→⋅p→$$+a$$→⋅q→$$+a$$→⋅r→$$+b$$→⋅p→$$+b$$→⋅q→$$+b$$→⋅r→$$+c$$→⋅p→$$+c$$→⋅q→$$+c$$→⋅r→$$=1+1+1=3$$

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p→=b→×c→[a→b→c→],q→=c→×a→[a→b→c→] and r→=a→×b→[a→b→c→] where a→,b→ and c→ are thee non-coplanar vectors, then the value of the expression (a→+b→+c→)⋅(p→+q→+r→) is

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