Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa

Let \mathrm{\vec{a}, \vec{b}, \vec{c}} be three vectors such that \mathrm{|\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2\: and \: 2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})}. If the angle between \mathrm{\vec{b} \: and\: \vec{c}\: is \: \frac{2 \pi}{3}}, then \mathrm{\left(\frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2} is equal to
 

Option: 1

3


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{ |\vec{a}|=\sqrt{31} \quad 4|\vec{b}|=|\overrightarrow{\mathrm{c}}|=2 }
\mathrm{ 2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})=3(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}) }
\mathrm{ \overrightarrow{\mathrm{b}} \wedge \overrightarrow{\mathrm{c}}=\frac{2 \pi}{3} }
\mathrm{ \overrightarrow{\mathrm{a}} \times 2 \overrightarrow{\mathrm{b}}=3 \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}=-\overrightarrow{\mathrm{a}} \times 3 \overrightarrow{\mathrm{c}} }
\mathrm{ \overrightarrow{\mathrm{a}} \times(2 \overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}})=\overrightarrow{\mathrm{o}} }
\mathrm{ \overrightarrow{\mathrm{a}}=\lambda(2 \overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}) }
\mathrm{ |\overrightarrow{\mathrm{a}}|^2=\lambda^2|2 \overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}|^2 }
\mathrm{ |\overrightarrow{\mathrm{a}}|^2=\lambda^2\left(4|\vec{b}|^2+9|\vec{c}|^2+12|\overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}| \cos \theta\right) }
\mathrm{ 31=\lambda^2\left(1+9(2)^2+12|\overrightarrow{\mathrm{b}} \| \overrightarrow{\mathrm{c}}| \cos \frac{2 \pi}{3}\right) }
\mathrm{ 31=\lambda^2\left(1+36-6 \times \frac{1}{2} \times 2\right) }
\mathrm{ 31=\lambda^2(31) }
\mathrm{ \lambda^2=1 }
\mathrm{ \Rightarrow \lambda= \pm 1 }
\mathrm{ \overrightarrow{\mathrm{a}}= \pm(2 \overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}) }

\mathrm{ \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}= \pm(2 \overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{c}} }
\mathrm{ = \pm 2(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}) }
\mathrm{ |\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}|^2=4|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|^2=3 }
\mathrm{ \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\mp 1 }
\mathrm{ \left(\frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}|}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right)^2=3 }

 

Posted by

Suraj Bhandari

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

BTech Admission 2025 Live: JEE Main correction window closes at 11:50 pm; VITEEE, SRMJEE, AEEE dates out
anam-khan

JEE Main 2025 application form correction begins at jeemain.nta.nic.in; editable fields
anam-khan

JEE Main 2025 application form correction window opens today; apply at jeemain.nta.nic.in

Latest Question