Let three vectors a→,b→ and c→ be such that c→ is coplanar with a→ and b→,a→⋅c→$$=7$$ and b→ is perpendicular to c→, where a→$$=$$−$$i^+j^+k^$$ and b→$$=$$−$$2i^+k^$$, then the value of $$2$$|a→$$+b$$→$$+c$$→|$$2$$ is

Question

Let three vectors a→,b→ and c→ be such that c→ is coplanar with a→ and b→,a→⋅c→$$=7$$ and b→ is  perpendicular to c→, where a→$$=$$−$$i^+j^+k^$$ and b→$$=$$−$$2i^+k^$$, then the value of $$2$$|a→$$+b$$→$$+c$$→|$$2$$ is

Infinity Learn · Accepted Answer

c→$$=$$λ$$(b$$→×$$(a$$→×b→$$))=$$λ$$((b$$→⋅b→$$)a$$→−$$(b$$→⋅a→$$)b$$→$$)=$$λ$$(5($$−$$i^+j^+k^)+2i^+k^)=$$λ$$($$−$$3i^+5j^+6k^)c$$→a→$$=7$$⇒$$3$$λ$$+5$$λ$$+6$$λ$$=7$$⇒λ$$=12$$∴$$2a$$→$$+b$$→$$+c$$→$$2=2$$−$$32$$−$$1+2i^+52+1j^+(3+1+1)k^$$                                 $$=214+494+25$$                                $$=25+50=75$$

Vector triple product

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Vector triple product

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Let three vectors a→,b→ and c→ be such that c→ is coplanar with a→ and b→,a→⋅c→=7 and b→ is perpendicular to c→, where a→=−i^+j^+k^ and b→=−2i^+k^, then the value of 2|a→+b→+c→|2 is

c→=λ(b→×(a→×b→))=λ((b→⋅b→)a→−(b→⋅a→)b→)=λ(5(−i^+j^+k^)+2i^+k^)=λ(−3i^+5j^+6k^)c→a→=7⇒3λ+5λ+6λ=7⇒λ=12∴2a→+b→+c→2=2−32−1+2i^+52+1j^+(3+1+1)k^ =214+494+25 =25+50=75

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