If a1→ and a2→ are two non collinear unit vectors and if a1→+a2→=3 then the value of a1→−a2→⋅2a1→+a2→ is:

a→1=a→2=1; If a→1+a→2=3⇒a12+a22+2a1a2cos⁡θ=3⇒cos⁡θ=1/2⇒θ=60∘ Now a→1−a→2⋅2a→1+a→2=2a12−2a→2⋅a→1+a→1⋅a→2−a22=2a12−a→1⋅a→2−a22=2−12−1=12

If a1→ and a2→ are two non collinear unit vectors and if a1→+a2→=3 then the value of a1→−a2→⋅2a1→+a2→ is:

a→1=a→2=1; If a→1+a→2=3⇒a12+a22+2a1a2cos⁡θ=3⇒cos⁡θ=1/2⇒θ=60∘ Now a→1−a→2⋅2a→1+a→2=2a12−2a→2⋅a→1+a→1⋅a→2−a22=2a12−a→1⋅a→2−a22=2−12−1=12

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If $\vec{a_{1}} and \vec{a_{2}}$ are two non collinear unit vectors and if $|\vec{a_{1}} + \vec{a_{2}}| = \sqrt{3}$ then the value of $(\vec{a_{1}} - \vec{a_{2}}) \cdot (2 \vec{a_{1}} + \vec{a_{2}})$ is:

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$|{\vec{a}}_{1}| = |{\vec{a}}_{2}| = 1; If |{\vec{a}}_{1} + {\vec{a}}_{2}| = \sqrt{3}$

$\Rightarrow \sqrt{a_{1}^{2} + a_{2}^{2} + 2 a_{1} a_{2} \cos θ} = \sqrt{3}$

$\Rightarrow \cos θ = 1 / 2 \Rightarrow θ = 60^{\circ}$

$Now ({\vec{a}}_{1} - {\vec{a}}_{2}) \cdot (2 {\vec{a}}_{1} + {\vec{a}}_{2})$

$= 2 a_{1}^{2} - 2 {\vec{a}}_{2} \cdot {\vec{a}}_{1} + {\vec{a}}_{1} \cdot {\vec{a}}_{2} - a_{2}^{2}$

$= 2 a_{1}^{2} - {\vec{a}}_{1} \cdot {\vec{a}}_{2} - a_{2}^{2} = 2 - \frac{1}{2} - 1 = \frac{1}{2}$

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If a1→ and a2→ are two non collinear unit vectors and if a1→+a2→=3 then the value of a1→−a2→⋅2a1→+a2→ is:

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If $\vec{a_{1}} and \vec{a_{2}}$ are two non collinear unit vectors and if $|\vec{a_{1}} + \vec{a_{2}}| = \sqrt{3}$ then the value of $(\vec{a_{1}} - \vec{a_{2}}) \cdot (2 \vec{a_{1}} + \vec{a_{2}})$ is: