A parallelogram is constructed on $$5a$$→$$+2b$$→ and a→−$$3b$$→ as adjacent sides, where |a→|$$=22$$ and |b→|$$=3$$ & angle between a→&b→ is π$$4$$ . If the magnitude of longer diagonal is αβγ where αβγ is three digit number then α$$+$$β−γ is equal to

Question

A parallelogram is constructed on $$5a$$→$$+2b$$→ and a→−$$3b$$→ as adjacent sides, where |a→|$$=22$$ and |b→|$$=3$$ & angle between a→&b→ is π$$4$$ . If the magnitude of longer diagonal is αβγ where αβγ is three digit number then α$$+$$β−γ is equal to

Infinity Learn · Accepted Answer

Given that |a→|$$=22$$,|b→|$$=3$$ and angle between a→ and b→ is π$$4$$ The diagonals of the parallelogram are p→$$=5a$$→$$+2b$$→$$+a$$→−$$3b$$→$$=6a$$→−b→ and q→$$=4a$$→$$+5b$$→|p→|$$=$$|$$6$$→a→−b→|$$=36$$|a→|$$2+$$|b→|$$2$$−$$12(a$$→⋅b→$$)=36$$×$$8+9$$−$$12$$×$$22$$×$$3$$×$$12=15$$ And |q→|$$=$$|$$4a$$→$$+5b$$→|$$=16$$|a→|$$2+25$$|b→|$$+40(a$$·→b→$$)=16$$×$$8+25$$×$$9+40$$×$$22$$×$$3$$×$$12=593$$∴ Magnitude of longer diagonal $$=593=$$αβγ (given) ∴α$$+$$β−γ$$=5+9$$−$$3=11$$

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A parallelogram is constructed on 5a→+2b→ and a→−3b→ as adjacent sides, where |a→|=22 and |b→|=3 & angle between a→&b→ is π4 . If the magnitude of longer diagonal is αβγ where αβγ is three digit number then α+β−γ is equal to

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