Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry3 Marks
Question
Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are
$\vec r = \hat i + \hat j + \lambda (2\hat i - \hat j + \hat k)$ ...(1)
and $\vec r = 2\hat i + \hat j - \hat k + \mu (3\hat i - 5\hat j + 2\hat k)$ ...(2)
✓

Answer

${\vec a_1} = \hat i + \hat j,{\vec b_1} = 2\hat i - \hat j + \hat k$
${\vec a_2} = 2\hat i + \hat j - \hat k,{\vec b_2} = 3\hat i - 5\hat j + 2\hat k$
${\vec a_2} - {\vec a_1} = \hat i - \hat k$
${\vec b_1} \times {\vec b_2} = \left| {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k} \\ 2&{ - 1}&1 \\ 3&{ - 5}&2 \end{array}} \right|$
$=\hat i (-2+5)-\hat j(4-3)+\hat k(-10+3)$
$ = 3\hat i - \hat j - 7\hat k$
$\left| {{{\vec b}_1} \times {{\vec b}_2}} \right|=\sqrt {9+1+49}= \sqrt {59} $
Also, $(\vec b_1×\vec b_2).(\vec a_2-\vec a_1)=(3\hat i-\hat j-7 \hat k)(\hat i-\hat k)=3+7+0=10$
$d=\left|\frac{(\vec b_1×\vec b_2).(\vec a_2-\vec a_1)}{|\vec b_1×\vec b_2|}\right|=\frac{10}{\sqrt{59}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove the following results:
$2\tan^{-1}\frac{3}{4}-\tan^{-1}\frac{17}{31}=\frac{\pi}{4}$
→
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\sin\text{x }\text{dx}$
→
Discuss the continuity of the function f(x) at the point x = 0, where$\text{f}\text{(x)}=\begin{cases}\text{x}, & \text{x} > 0\\1,&\text{x}=0\\\text{-x}, & \text{x} > 0\end{cases}$
→
By using the properties of definite integrals, evaluate the integral in Exercise:
$\int^{\frac{\pi}{2}}_{0}\big(2\log\sin\text{x}-\log\sin2\text{x}\big)\text{dx}$
→
Write the projection of $\vec{\text{b}}+\vec{\text{c}}$ on $\vec{\text{a}}$ when $\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$
→
Two cards are drawn successively without replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
→
If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
→
If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two vectors of the same magnitude inclined at an angle of 30°, such that $\vec{\text{a}}.\vec{\text{b}}=3,$ find $|\vec{\text{a}}|,\big|\vec{\text{b}}\big|.$
→
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and, hence, find its mean.
→
A factory has two machines A and B. Past records show that the machine A produced $60 \%$ of the items of output and machine B produced $40 \%$ of the items. Further $2 \%$ of the items produced by machine A were defective and $1 \%$ produced by machine B were defective. If an item is drawn at random, what is the probability that it is defective?
→