Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Let   be a non-zero vector parallel to the line of intersection of the two places described by&nbs

Let \bar{a}  be a non-zero vector parallel to the line of intersection of the two places described by \hat{\mathrm{i}}+\hat{\mathrm{j}}, \hat{\mathrm{i}}+\hat{\mathrm{k}}and \hat{\mathrm{i}}+\hat{\mathrm{j}}, \hat{\mathrm{i}}-\hat{\mathrm{k}} if \theta is the angle between the vector \bar{a} and the vector\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \text { and } \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=6 \text {, }  then ordered pair (\theta,|\vec{a} \times \vec{b}|) is equal to

Option: 1

\left(\frac{\pi}{3}, 6\right)


Option: 2

\left(\frac{\pi}{4}, 3 \sqrt{6}\right)


Option: 3

\left(\frac{\pi}{3}, 3 \sqrt{6}\right)


Option: 4

\left(\frac{\pi}{4}, 6\right)


Answers (1)

best_answer

\overrightarrow{\mathrm{n}}_1and \overrightarrow{\mathrm{n}}_2 are normal vector to the plane \hat{\mathrm{i}}+\hat{\mathrm{j}}, \hat{\mathrm{i}}+\hat{\mathrm{k}} and \hat{\mathrm{i}}-\hat{\mathrm{j}}, \hat{\mathrm{i}}-\hat{\mathrm{k}} respectively 00^{\circ} 0 N^{\circ} \begin{aligned} & \overrightarrow{\mathrm{n}}_1=\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{j}} \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right|=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{n}}_2=\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{j}} \\ 1 & -1 & 0 \\ 1 & 0 & -1 \end{array}\right|=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}} \end{aligned} \overrightarrow{\mathrm{a}}=\lambda\left|\overrightarrow{\mathrm{n}}_2 \times \overrightarrow{\mathrm{n}}_2\right|

$$ \begin{aligned} & =\lambda\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{j}} \\ 1 & -1 & -1 \\ 1 & 1 & -1 \end{array}\right|=\lambda(-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \\ & \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\lambda|0+4+2|=6 \\ & \Rightarrow \lambda=1 \\ & \vec{\alpha}=-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}} \\ & \cos \theta=\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\mathrm{a}||\mathrm{b}|} \\ & \cos \theta=\frac{6}{2 \sqrt{2} \times 3}=\frac{1}{\sqrt{2}} \\ & \theta=\frac{\pi}{4} \end{aligned} $$ Now |\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|^2+|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^2=|\mathrm{a}|^2|\mathrm{~b}|^2 \begin{aligned} & 36+\left|\overrightarrow{\mathrm{a}} \times \mathrm{b}^2\right|=8 \times 9=72 \\ & |\overrightarrow{\mathrm{a}} \times \mathrm{b}|^2=36 \\ & |\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=6 \end{aligned}

Posted by

qnaprep

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

JEE Main 2025: 11 from Kota coaching centres among 24 students with 100 percentile
anam-khan

Low JEE Main rank? Explore alternative paths for engineering admission
anam-khan

Gender-gap in JEE Main 2025: Female participation remains around 31%, overall pool drops by 7.09%

Latest Question