If a→ and c→ are unit vectors and |b→|=5.The angle between a→ and c→ is π3and b→−2c→=αa→ then α2+2α+10=

b→−2c→=αa→⇒b→=2c→+αa→⇒|b→|2=|2c→+αa→|2=4|c→|2+α2|a→|2+4αa→⋅c→⇒5=4+α2+(4α)12α2+2α−1=0

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If $\vec{a}$ and $\vec{c}$ are unit vectors and $| \vec{b} | = \sqrt{5} .$ The angle between $\vec{a}$ and $\vec{c}$ is $\frac{π}{3}$ and $\vec{b} - 2 \vec{c} = α \vec{a}$ then $α^{2} + 2 α + 10 =$

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Detailed Solution

$\begin{array}{l} \vec{b} - 2 \vec{c} = α \vec{a} \Rightarrow \vec{b} = 2 \vec{c} + α \vec{a} \\ \Rightarrow | \vec{b} |^{2} = | 2 \vec{c} + α \vec{a} |^{2} = 4 | \vec{c} |^{2} + α^{2} | \vec{a} |^{2} + 4 α \vec{a} \cdot \vec{c} \\ \Rightarrow 5 = 4 + α^{2} + (4 α) (\frac{1}{2}) \\ α^{2} + 2 α - 1 = 0 \end{array}$