Question

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Answer 1

QUESTION :–

• If $\bf \overrightarrow{a} \: , \: \overrightarrow{b} \: \: and \: \: \overrightarrow{c}$ be three vectors such that $\bf \overrightarrow{a} \: + \: \overrightarrow{b} \:+\: \overrightarrow{c} = 0$ and $\bf | \overrightarrow{a} | = 3 \:, \: | \overrightarrow{b} | = 5 \: , \: | \overrightarrow{c} | = 7$ , find the angel between $\bf \overrightarrow{a} \: \: and \: \: \overrightarrow{b}.$

ANSWER :–

GIVEN :–

$\\ \bf \: \: {\huge{.}} \: \: \: \overrightarrow{a} \: + \: \overrightarrow{b} \:+\: \overrightarrow{c} = 0$

$\\ \bf \: \: {\huge{.}} \: \: \: | \overrightarrow{a} | = 3 \:, \: | \overrightarrow{b} | = 5 \: , \: | \overrightarrow{c} | = 7$

TO FIND :–

• Angle between $\bf \overrightarrow{a} \: \: and \: \: \overrightarrow{b} = ?$

SOLUTION :–

$\\ \bf \implies \overrightarrow{a} \: + \: \overrightarrow{b} \:+\: \overrightarrow{c} = 0$

• We should write this as –

$\\ \bf \implies \overrightarrow{a} \: + \: \overrightarrow{b} \: = - \: \overrightarrow{c}$

• Now square on both sides –

$\\ \bf \implies( \overrightarrow{a} \: + \: \overrightarrow{b})^{2} \: = ( - \: \overrightarrow{c})^{2}$

$\\ \bf \implies | \overrightarrow{a} |^{2}+ |\overrightarrow{b}|^{2} + 2(\overrightarrow{a}.\overrightarrow{b})= | \overrightarrow{c}|^{2}$

• Using identity –

$\\ \large \implies{ \boxed{ \bf \overrightarrow{a}.\overrightarrow{b} = | \overrightarrow{a} | | \overrightarrow{b} | \cos( \theta)}}$

$\\ \bf \implies | \overrightarrow{a} |^{2}+ |\overrightarrow{b}|^{2} + 2| \overrightarrow{a} | | \overrightarrow{b} | \cos( \theta)= | \overrightarrow{c}|^{2}$

• Now put the values –

$\\ \bf \implies (3)^{2}+(5)^{2} + 2(3)(5)\cos( \theta)=(7)^{2}$

$\\ \bf \implies 9+25+30\cos( \theta)=49$

$\\ \bf \implies 34+30\cos( \theta)=49$

$\\ \bf \implies 30\cos( \theta)=49 - 34$

$\\ \bf \implies 30\cos( \theta)=15$

$\\ \bf \implies \cos( \theta)= \cancel \dfrac{15}{30}$

$\\ \bf \implies \cos( \theta)=\dfrac{1}{2}$

$\\ \bf \implies \cos( \theta)= \cos( {60}^{ \circ} )$

$\\ \large\implies{ \boxed{ \bf \theta={60}^{ \circ}}}$

▪︎ Hence, Angle between $\bf \overrightarrow{a} \: \: and \: \: \overrightarrow{b}$ is 60° .

Answer 2

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