If a → = 2 , b → = 3 and 2 a → - b → = 5 , then 2 a → + b → equals :

a → = 2 , b → = 3 , 2 a → - b → = 5 2 a → - b → 2 = 2 a → - b → · 2 a → - b → = 4 a → 2 - 4 a → · b → + b → 2 . . . i 2 a → + b → 2 = 4 a → 2 + 4 a → · b → + b → 2 . . . ii Adding equation (i) and equation (ii), we get 2 a → + b → 2 + 2 a → - b → 2 = 8 a → 2 + 2 b → 2 ⇒ 2 a → + b → 2 + 5 2 = 8 2 2 + 2 3 2 ⇒ 2 a → + b → 2 = 32 + 18 - 25 = 2 5 ⇒ 2 a → + b → = 5

Mathematics - Vector Question with Solution | TestHub

MathematicsVectorDot Product & Its Application ( Projection etc.)Easy2 minPYQ_2014

MathematicsEasysingle choice

If $|\vec{a}| = 2, |\vec{b}| = 3$ and $|2 \vec{a} - \vec{b}| = 5,$ then $|2 \vec{a} + \vec{b}|$ equals :

Options:

Answer:

Solution:

$|\vec{a}| = 2, |\vec{b}| = 3, |2 \vec{a} - \vec{b}| = 5$

${|2 \vec{a} - \vec{b}|}^{2} = (2 \vec{a} - \vec{b}) \cdot (2 \vec{a} - \vec{b})$

$= 4 {|\vec{a}|}^{2} - 4 [\vec{a} \cdot \vec{b}] + {|\vec{b}|}^{2} . . . (i)$

${|2 \vec{a} + \vec{b}|}^{2} = 4 {|\vec{a}|}^{2} + 4 [\vec{a} \cdot \vec{b}] + {|\vec{b}|}^{2} . . . (ii)$

Adding equation (i) and equation (ii), we get

${|2 \vec{a} + \vec{b}|}^{2} + {|2 \vec{a} - \vec{b}|}^{2} = 8 {|\vec{a}|}^{2} + {2 |\vec{b}|}^{2}$

$\Rightarrow {|2 \vec{a} + \vec{b}|}^{2} + 5^{2} = 8 (2^{2}) + 2 (3^{2})$

$\Rightarrow {|2 \vec{a} + \vec{b}|}^{2} = 32 + 18 - 25$ $= 25$

$\Rightarrow |2 \vec{a} + \vec{b}| = 5$

Stream:JEESubject:MathematicsTopic:VectorSubtopic:Dot Product & Its Application ( Projection etc.)

⏱ 2mℹ️ Source: PYQ_2014

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