Question

6. If[a] = 13,[b]= 5 and a.b = 60, then find la xbl.
How​

Answer 1

$\textbf{Formula used:}$

$\text{For any two vectors \overrightarrow{a} and \overrightarrow{b} ,}$

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

$\textbf{Given:}$

$|\overrightarrow{a}|=13$

$|\overrightarrow{b}|=5$

$\overrightarrow{a}.\overrightarrow{b}=60$

$\text{Now,}$

$|\overrightarrow{a}{\times}\overrightarrow{b}|^2+(\overrightarrow{a}.\overrightarrow{b})^2=|\overrightarrow{a}|^2\,|\overrightarrow{b}|^2$

$|\overrightarrow{a}{\times}\overrightarrow{b}|^2+(60)^2=(13)^2\,(5)^2$

$|\overrightarrow{a}{\times}\overrightarrow{b}|^2+(60)^2=(65)^2$

$|\overrightarrow{a}{\times}\overrightarrow{b}|^2=(65)^2-(60)^2$

$|\overrightarrow{a}{\times}\overrightarrow{b}|^2=(65+60)(65-60)$

$|\overrightarrow{a}{\times}\overrightarrow{b}|^2=(125)(5)$

$|\overrightarrow{a}{\times}\overrightarrow{b}|^2=625$

$\implies|\overrightarrow{a}{\times}\overrightarrow{b}|=\sqrt{625}$

$\implies\bf|\overrightarrow{a}{\times}\overrightarrow{b}|=25$

$\therefore\textbf{The value of |\overrightarrow{a}{\times}\overrightarrow{b}| is 25}$

Answer 2

GIVEN :–

$\\ \bf \implies | \overrightarrow{a}| = 3 \: , \:| \overrightarrow{b}| =4\: , \: \overrightarrow{a}.\overrightarrow{b} = 9$

TO FIND :–

• Value of $\bf \overrightarrow{a} \times \overrightarrow{b} =?$

SOLUTION :–

$\\ \bf \implies\overrightarrow{a}.\overrightarrow{b} = 9$

$\\ \bf \implies | \overrightarrow{a}| | \overrightarrow{b}| \cos( \theta) = 9$

$\\ \bf \implies (3)(4)\cos( \theta) = 9$

$\\ \bf \implies \cos( \theta) = \dfrac{9}{12}$

$\\ \bf \implies \cos(\theta) = \dfrac{3}{4}$

• We should write this as –

$\\ \bf \implies \sqrt{1 - { \sin}^{2}(\theta)} = \dfrac{3}{4}$

• Square on both side –

$\\ \bf \implies 1 - { \sin}^{2}(\theta)= \dfrac{9}{16}$

$\\ \bf \implies { \sin}^{2}(\theta)= 1 - \dfrac{9}{16}$

$\\ \bf \implies { \sin}^{2}(\theta)= \dfrac{16 - 9}{16}$

$\\ \bf \implies { \sin}^{2}(\theta)= \dfrac{7}{16}$

$\\ \large \implies{ \boxed{ \bf\sin(\theta)= \dfrac{ \sqrt{7}}{4}}}$

• Now let's find –

$\\ \implies \bf\overrightarrow{a} \times \overrightarrow{b} =| \overrightarrow{a}| | \overrightarrow{b}| \sin(\theta)$

$\\ \implies \bf\overrightarrow{a} \times \overrightarrow{b} =(3)(4)\dfrac{ \sqrt{7}}{4}$

$\\ \large \implies \red{ \boxed{\bf\overrightarrow{a} \times \overrightarrow{b} =3\sqrt{7}}}$