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  • Let <img alt="\mathrm{\overrightarrow{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}}" src="/latex-image/?%5Cmathrm%7B%5Coverrightarrow%7Ba%7D%3D%5Calpha%20%5Chat%7Bi%7D&plus;%5Chat%7Bj%7D&pl

Let \mathrm{\overrightarrow{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}}  and \mathrm{\overrightarrow{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}} be two vectors, such that  \mathrm{\overrightarrow{a} \times \overrightarrow{b}=-\hat{i}+9 \hat{i}+12 \hat{k}}.. Then the projection of \mathrm{\overrightarrow{b}-2 \overrightarrow{a}} on \mathrm{\overrightarrow{b}+\overrightarrow{a}} is equal to 

Option: 1

2


Option: 2

\frac{39}{5}


Option: 3

9


Option: 4

\frac{46}{5}


Answers (1)

best_answer

\mathrm{\text { Let } \overrightarrow{a}=\alpha \hat{i}+\hat{j}+\beta k, \overrightarrow{b}=3 \hat{i}-5 f+4 \hat{k}} \\

\mathrm{\overrightarrow{a} \times \overrightarrow{b}=-\hat{i}+9 \hat{j}+12 \hat{k}} \\

\mathrm{\Rightarrow\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \alpha & 1 & \beta \\ 3 & -5 & 4 \end{array}\right| }\\

\mathrm{\Rightarrow(4+5 \beta) \hat{i}+(3 \beta-4 \alpha) \hat{j}+(-5 \alpha-3) \hat{k} }\\

\mathrm{=-\hat{i}+9 \hat{j}+12 \hat{k}}

\mathrm{\therefore 4+5 \beta=+, 3 \beta-4 \alpha=9,-5 \alpha-3=12} \\

\mathrm{\therefore {\beta }=-1, \alpha=-3} . \\

\mathrm{\therefore \overrightarrow{\alpha }=-3 \hat{i}+\hat{j}-\hat{k}, \overrightarrow{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}} \\

\mathrm{\therefore \overrightarrow{a}+\overrightarrow{b}=-4 \hat{j}+3 \hat{k}} \\

\mathrm{|\overrightarrow{a}|^{2}=11,|\overrightarrow{b}|^{2}=50 }\\

\mathrm{\overrightarrow{a} \cdot \overrightarrow{b}=-9+(-5)-4=-18}

\mathrm{\therefore Projectile \;of (\overrightarrow{b}-2 \overrightarrow{a}) on \overrightarrow{a}+\overrightarrow{b} f}

\mathrm{ \frac{(\overrightarrow{b}-2 \overrightarrow{a}) \cdot(\overrightarrow{a}+\overrightarrow{b})}{|\overrightarrow{a}+\overrightarrow{b}|}=\frac{\left.|\overrightarrow{b}|^{2}-2|\overrightarrow{a}|^{2}-\mid \overrightarrow{a} \cdot \overrightarrow{b}\right)}{|\overrightarrow{a}+\overrightarrow{b}|}} \\

\mathrm{= \frac{50-22-(-18)}{-5}=\frac{46}{5} }

Hence correct option is 4

 

Posted by

Nehul

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