logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  CBSE XII  >>  Math  >>  Three Dimensional Geometry
0 votes

$\overrightarrow{AB}=3 \hat i-\hat j+\hat k \;and\; \overrightarrow{CD}=-3\hat i+2 \hat j+4\hat k$ are two vectors. The position vectors of the points $A$ and $C$ are $6\hat i+7\hat j+4\hat k\;and\;-9\hat j+2\hat k,$ respectively.Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{PQ}$ is perpendicular to $\overrightarrow{AB}\;and\;\overrightarrow{CD}$ both.

$\begin{array}{1 1} P(3\hat i + 8\hat j + 3\hat k) \: and \: Q(-3\hat i - 7\hat j + 6\hat k ) \\ P(2\hat i + 8\hat j + 3\hat k) \: and \: Q(-2\hat i - 7\hat j + 6\hat k )\\ P(3\hat i + 7\hat j + 3\hat k) \: and \: Q(-3\hat i +7\hat j + 6\hat k ) \\P(3\hat i + 8\hat j - 3\hat k) \: and \: Q(3\hat i - 7\hat j + 6\hat k ) \end{array} $

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • Write the cartesian equations of the lines $ \overline{AB}\: and \: \overline{CD}$
  • Two lines $ \perp$ means dot product of thier d.r = 0
  • Position vector of point is coordinates of the point.
Step 1
Write the equation of AB and CD in cartesian form.
$ AB \rightarrow \large\frac{x-6}{3}=\large\frac{y-7}{-1}=\large\frac{z-4}{1}=\lambda $( say)
$ CD \rightarrow \large\frac{x+0}{-3}=\large\frac{y+9}{2}=\large\frac{z-2}{4}=\mu $( say)
Step 2
Point p on $ \overline{AB} $ and point Q on CD are given by
$ p(3\lambda+6, -\lambda+7, \lambda+4)$
$ Q(-3\mu, 2\mu-9, 4\mu+2)$
Step 3
$ \overline{PQ}=(3\lambda+3\mu+6,-\lambda-24+16, \lambda-4\mu+2) $
Step 4
given $ \overline{PQ} $ is perpendicular to
$ AB\: and \: CD \: \Rightarrow \overrightarrow{PQ}.\overrightarrow{AB}=0\: and \: \overrightarrow{PQ}.\overrightarrow{CD}=0$
$ \Rightarrow 11\lambda+7\mu+4=0\: and \: -7\mu-29\mu+22=0$
Step 5
Solving which $ \lambda $=
$ \mu =$
Step 6
Put the values of $ \lambda \: and \: \mu$ in P and Q and get the required vector
$ P(3\hat i + 8\hat j + 3\hat k) \: and \: Q(-3\hat i - 7\hat j + 6\hat k )$

 

answered Mar 7, 2013 by thanvigandhi_1
edited Apr 5, 2013 by thanvigandhi_1
 

Related questions

Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...