Login/Sign Up
Home
If\,\left| {{A}_{1}} \right|=3,\,\left| {{A}_{2}} \right|=4\,and\,\left| {{A}_{1}}+{{A}_{2}} \right|=4,\,the\,value\,of\,\left( 2{{A}_{1}}+{{A}_{2}} \right).\left( {{A}_{1}}-{{A}_{2}} \right)\,is
MHTCET | Physics
If\,\left| {{A}_{1}} \right|=3,\,\left| {{A}_{2}} \right|=4\,and\,\left| {{A}_{1}}+{{A}_{2}} \right|=4,\,the\,value\,of\,\left( 2{{A}_{1}}+{{A}_{2}} \right).\left( {{A}_{1}}-{{A}_{2}} \right)\,is
  1. 4.5
  2. 5.5
  3. 6.5
  4. 2.5

Solution: 6.5

\left( 2\overrightarrow{{{A}_{1}}}+\overrightarrow{{{A}_{2}}} \right).\left( \overrightarrow{{{A}_{1}}}-\overrightarrow{{{A}_{2}}} \right)

Upon\,multiplying\,we\,get:

=2\overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{1}}}-2\overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{2}}}+\overrightarrow{{{A}_{2}}}.\overrightarrow{{{A}_{1}}}-\overrightarrow{{{A}_{2}}}.\overrightarrow{{{A}_{2}}}

=2{{\left| \overrightarrow{{{A}_{1}}} \right|}^{2}}-\overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{2}}}-{{\left| \overrightarrow{{{A}_{2}}} \right|}^{2}}

=2\times 9-\overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{2}}}-16

\left( 2\overrightarrow{{{A}_{1}}}+\overrightarrow{{{A}_{2}}} \right).\left( \overrightarrow{{{A}_{1}}}-\overrightarrow{{{A}_{2}}} \right)=2-\overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(i)

Also,\,\left| \overrightarrow{{{A}_{1}}}+\overrightarrow{{{A}_{2}}} \right|=4

Squaring\,both\,sides,\,we\,get:

\left( \overrightarrow{{{A}_{1}}}+\overrightarrow{{{A}_{2}}} \right).\left( \overrightarrow{{{A}_{1}}}+\overrightarrow{{{A}_{2}}} \right)=16

{{\left| \overrightarrow{{{A}_{1}}} \right|}^{2}}+2\left( \overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{2}}} \right)+{{\left| \overrightarrow{{{A}_{2}}} \right|}^{2}}=16

Putting\,known\,values,\,we\,get:

2\left( \overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{2}}} \right)=16-9-16

\Rightarrow \overrightarrow{{{A}_{1}}}.\overrightarrow{{{A}_{2}}}=-4.5

Putting\,this\,value\,in\,eq\,(i);

\left( 2\overrightarrow{{{A}_{1}}}+\overrightarrow{{{A}_{2}}} \right).\left( \overrightarrow{{{A}_{1}}}-\overrightarrow{{{A}_{2}}} \right)=2-\left( -4.5 \right)

\left( 2\overrightarrow{{{A}_{1}}}+\overrightarrow{{{A}_{2}}} \right).\left( \overrightarrow{{{A}_{1}}}-\overrightarrow{{{A}_{2}}} \right)=6.5

i

More Material