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$\left| {{{\vec A}_1}} \right| = 3,\,\left| {\vec A_2} \right| = 5$, અને $\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = 5$ આપેલ છે. $\left( {2{{\vec A}_1} + 3{{\vec A}_2}} \right)\cdot \left( {3{{\vec A}_1} - 2{{\vec A}_2}} \right)$ નું મૂલ્ય કેટલું થાય?
$-106.5$
$-112.5$
$-118.5$
$-99.5$
Solution
$\begin{array}{l}
\left| {{{\vec A}_1}} \right| = 3,\left| {{{\vec A}_2}} \right| = 5,\,and\,\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = 5.\\
\left| {{{\vec A}_1} + {{\vec A}_2}} \right| = {\left| {{{\vec A}_1}} \right|^2} + {\left| {{{\vec A}_2}} \right|^2} + 2\left| {{{\vec A}_1}} \right|\left| {{{\vec A}_2}} \right|\,\cos \,\theta \\
\cos \,\theta \, = – \frac{3}{{10}}\\
\left( {2{{\vec A}_1} + 3{{\vec A}_2}} \right).\left( {3{{\vec A}_1} – 2{{\vec A}_2}} \right)\\
= 6{\left| {{{\vec A}_1}} \right|^2} + 9{{\vec A}_1}.{{\vec A}_2} – 4{{\vec A}_1}.{{\vec A}_2} – 6{\left| {{{\vec A}_2}} \right|^2}\\
= – 118.5
\end{array}$
Similar Questions
જો $\left| {\vec A } \right|\, = \,2$ અને $\left| {\vec B } \right|\, = \,4$ હોય, તો કોલમ $-II$માં આપેલા ખૂણાને અનુરૂપ કોલમ $-I$માં આપેલા યોગ્ય સંબંધ સાથે જોડો.
| કોલમ $-I$ | કોલમ $-II$ |
| $(a)$ $\vec A \,.\,\,\vec B \, = \,\,0$ | $(i)$ $\theta = \,{0^o}$ |
| $(b)$ $\vec A \,.\,\,\vec B \, = \,\,+8$ | $(ii)$ $\theta = \,{90^o}$ |
| $(c)$ $\vec A \,.\,\,\vec B \, = \,\,4$ | $(iii)$ $\theta = \,{180^o}$ |
| $(d)$ $\vec A \,.\,\,\vec B \, = \,\,-8$ | $(iv)$ $\theta = \,{60^o}$ |
