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QUESTION :–
• If [tex] \bf \overrightarrow{a} \: , \: \overrightarrow{b} \: \: and \: \: \overrightarrow{c} [/tex] be three vectors such that [tex] \bf \overrightarrow{a} \: + \: \overrightarrow{b} \:+\: \overrightarrow{c} = 0 [/tex] and [tex] \bf | \overrightarrow{a} | = 3 \:, \: | \overrightarrow{b} | = 5 \: , \: | \overrightarrow{c} | = 7[/tex] , find the angel between [tex] \bf \overrightarrow{a} \: \: and \: \: \overrightarrow{b}.[/tex]
ANSWER :–
GIVEN :–
[tex] \\ \bf \: \: {\huge{.}} \: \: \: \overrightarrow{a} \: + \: \overrightarrow{b} \:+\: \overrightarrow{c} = 0 \\ [/tex]
[tex] \\ \bf \: \: {\huge{.}} \: \: \: | \overrightarrow{a} | = 3 \:, \: | \overrightarrow{b} | = 5 \: , \: | \overrightarrow{c} | = 7\\ [/tex]
TO FIND :–
• Angle between [tex] \bf \overrightarrow{a} \: \: and \: \: \overrightarrow{b} = ?[/tex]
SOLUTION :–
[tex] \\ \bf \implies \overrightarrow{a} \: + \: \overrightarrow{b} \:+\: \overrightarrow{c} = 0 \\ [/tex]
• We should write this as –
[tex] \\ \bf \implies \overrightarrow{a} \: + \: \overrightarrow{b} \: = - \: \overrightarrow{c} \\ [/tex]
• Now square on both sides –
[tex] \\ \bf \implies( \overrightarrow{a} \: + \: \overrightarrow{b})^{2} \: = ( - \: \overrightarrow{c})^{2} \\ [/tex]
[tex] \\ \bf \implies | \overrightarrow{a} |^{2}+ |\overrightarrow{b}|^{2} + 2(\overrightarrow{a}.\overrightarrow{b})= | \overrightarrow{c}|^{2} \\ [/tex]
• Using identity –
[tex] \\ \large \implies{ \boxed{ \bf \overrightarrow{a}.\overrightarrow{b} = | \overrightarrow{a} | | \overrightarrow{b} | \cos( \theta)}} \\ [/tex]
[tex] \\ \bf \implies | \overrightarrow{a} |^{2}+ |\overrightarrow{b}|^{2} + 2| \overrightarrow{a} | | \overrightarrow{b} | \cos( \theta)= | \overrightarrow{c}|^{2} \\[/tex]
• Now put the values –
[tex] \\ \bf \implies (3)^{2}+(5)^{2} + 2(3)(5)\cos( \theta)=(7)^{2} \\[/tex]
[tex] \\ \bf \implies 9+25+30\cos( \theta)=49\\[/tex]
[tex] \\ \bf \implies 34+30\cos( \theta)=49\\[/tex]
[tex] \\ \bf \implies 30\cos( \theta)=49 - 34\\[/tex]
[tex] \\ \bf \implies 30\cos( \theta)=15\\[/tex]
[tex] \\ \bf \implies \cos( \theta)= \cancel \dfrac{15}{30}\\[/tex]
[tex] \\ \bf \implies \cos( \theta)=\dfrac{1}{2}\\[/tex]
[tex] \\ \bf \implies \cos( \theta)= \cos( {60}^{ \circ} ) \\[/tex]
[tex] \\ \large\implies{ \boxed{ \bf \theta={60}^{ \circ}}}\\[/tex]
▪︎ Hence, Angle between [tex] \bf \overrightarrow{a} \: \: and \: \: \overrightarrow{b} [/tex] is 60° .
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