Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Let O be the origin and the position vector of the point P be  <img alt="-i\hat{}-2j\hat{}+3k\hat{}" src="https://entrancecorner.oncodecogs.com/gif.latex?-i%5Chat%7B%7D-2j%5Chat%7B%7D&amp

Let O be the origin and the position vector of the point P be  -i\hat{}-2j\hat{}+3k\hat{} . If the position vectors of the A, B and C are -2i\hat{}+j\hat{}-3k\hat{} , 2i\hat{}+4j\hat{}-2k\hat{} \, \, and  -4i\hat{}+2j\hat{}-k\hat{} respectively, then the projection of the vector OP on a vector  \begin{aligned} \overrightarrow{\mathrm{OP}} \end{aligned}perpendicular to the vectors     \begin{aligned} \overrightarrow{\mathrm{AB}} \end{aligned}   and   \begin{aligned} \overrightarrow{\mathrm{AC}} \end{aligned}   is :

Option: 1

\frac{10}{3}


Option: 2

\frac{8}{3}


Option: 3

\frac{7}{3}


Option: 4

3


Answers (1)

Position vector of the point P( -1, -2,3), A( -2,1, -3) B(2,4, -2), and C( -4,2, -1) 

Then      \overrightarrow{\mathrm{OP}} \cdot \frac{\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}}{\mid \overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}) \mid}

 

\begin{aligned} & \overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}=\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{j} & \hat{k} \\ 4 & 3 & 1 \\ -2 & 1 & 2 \end{array}\right| \\ & =\hat{\mathrm{i}}(5)-\hat{j}(8+2)+\hat{k}(4+6) \\ & =5 \hat{\mathrm{i}}-10 \hat{j}+10 \hat{k} \end{aligned}

Now 

\begin{aligned} & \overrightarrow{\mathrm{OP}} \cdot \frac{\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}}{|(\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}})|}=(-\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \cdot \frac{(5 \hat{\mathrm{i}}-10 \hat{\mathrm{j}}+10 \hat{k})}{\sqrt{(5)^2+(-10)^2+(10)^2}} \\ & =\frac{-5+20+30}{\sqrt{25+100+100}} \\ & =\frac{45}{\sqrt{225}}=\frac{45}{15}=3 \end{aligned}

 

 

 

Posted by

Sumit Saini

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: BTech CSE cut-offs for PwD candidates at NITs make entry easy; top institutes still selective
anam-khan

73 or 99 percentile? JEE Main result questioned, J-K candidate says, ‘Don’t create hype’
anam-khan

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

Latest Question