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  • Let <img alt="\overrightarrow{\mathrm{a}}=\hat{i}+5 \hat{j}+\alpha \hat{k}, \vec{b}=\hat{i}+3 \hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2 \hat{j}-3 \hat{k}" src="https://entrancecorner.codecogs.com/gif.latex?%5Coverrightarrow%7B%5Cmathrm%7Ba%7D%7D%3D%5

Let \overrightarrow{\mathrm{a}}=\hat{i}+5 \hat{j}+\alpha \hat{k}, \vec{b}=\hat{i}+3 \hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2 \hat{j}-3 \hat{k} be three vectors such that, |\vec{b} \times \vec{c}|=5 \sqrt{3}$ and $\vec{a} is perpendicular to \vec{b}. Then the greatest amongst the values of |\vec{a}|^{2} is__________
 

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\vec{b}\times \vec{c}= \begin{vmatrix} \hat{i} & \hat{j} &\hat{k} \\ 1& 3 &\beta \\ -1& 2 & -3 \end{vmatrix}
= \left ( -9-2\beta \right )\hat{i}+\left ( 3-\beta \right )\hat{j}+5\hat{k}
\left | \vec{b}\times \vec{c} \right |^{2}= 75
\Rightarrow 4\beta ^{2}+36\beta +81+\beta ^{2}-6\beta +9+25= 75
\Rightarrow 5\beta ^{2}+30\beta +40= 0
\Rightarrow 5\beta ^{2}+20\beta +10\beta +40= 0
\Rightarrow \beta = -4 \: \: OR\, \,2
\vec{a}\cdot \vec{b}= 0\Rightarrow 1+15+\alpha \beta = 0
\Rightarrow \alpha \beta = -16
if\, \beta = -4;\alpha = 4,if \: \beta = -2,\alpha = 8
\left | \vec{a}\right |^{2}_{max}= 1^{2}+5^{2}+8^{2}
            =90

Posted by

Kuldeep Maurya

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