Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry5 Marks
Question
By computing the shortest distance determine whether the following pairs of lines intersect or not:
$\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}-\hat{\text{j}}\big)$ and $\vec{\text{r}}=\big(4\hat{\text{i}}-\hat{\text{k}}\big)+\mu\big(2\hat{\text{i}}+3\hat{\text{k}}\big)$
✓

Answer

Given equations of lines are,
$\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}-\hat{\text{j}}\big)$
$\Rightarrow\vec{\text{a}}_1=\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big),\vec{\text{b}}_1=\big(3\hat{\text{i}}-\hat{\text{j}}\big)$
and, $\vec{\text{r}}=\big(4\hat{\text{i}}-\hat{\text{k}}\big)+\mu\big(2\hat{\text{i}}+3\hat{\text{k}}\big)$
$\Rightarrow\vec{\text{a}}_2=\big(4\hat{\text{i}}-\hat{\text{k}}\big),\vec{\text{b}}_2=\big(2\hat{\text{i}}+3\hat{\text{k}}\big)$
We know that, shortest distance between lines $\vec{\text{r}}=\vec{\text{a}}_1+\lambda\vec{\text{b}}_1$ and $\vec{\text{r}}=\vec{\text{a}}_2+\lambda\vec{\text{b}}_2$ is given by
$\text{S.D.}=\Bigg|\frac{\big(\vec{\text{a}}_2-\vec{\text{a}}_1\big).\big(\vec{\text{b}}_1\times\vec{\text{b}}_2\big)}{\big|\vec{\text{b}}_1\times\vec{\text{b}}_2\big|}\Bigg|\dots(1)$
$\big(\vec{\text{a}}_2-\vec{\text{a}}_1\big)=\big(4\hat{\text{i}}-\hat{\text{k}}\big)-\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$
$=4\hat{\text{i}}-\hat{\text{k}}-\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
$=3\hat{\text{i}}-\hat{\text{j}}$
$\vec{\text{b}}_1\times\vec{\text{b}}_2=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\3&-1&0\\2&0&3 \end{vmatrix}$
$=\hat{\text{i}}(-3-0)-\hat{\text{j}}(9-0)+\hat{\text{k}}(0+2)$
$=-3\hat{\text{i}}-9\hat{\text{j}}+2\hat{\text{k}}$
$\big|\vec{\text{b}}_1\times\vec{\text{b}}_2\big|=\sqrt{(-3)^2+(-9)^2+(2)^2}$
$\big|\vec{\text{b}}_1\times\vec{\text{b}}_2\big|=\sqrt{9+81+4}$
$=\sqrt{94}$
$\big(\vec{\text{a}}_2-\vec{\text{a}}_1\big).\big(\vec{\text{b}}_1\times\vec{\text{b}}_2\big)=\big(3\hat{\text{i}}-\hat{\text{j}}\big)\big(-3\hat{\text{i}}-9\hat{\text{j}}+2\hat{\text{k}}\big)$
$=(3)(-3)+(-1)(-9)+(0)(2)$
$=-9+9+0$
$=0$
Using $\big(\vec{\text{a}}_2-\vec{\text{a}}_1\big).\big(\vec{\text{b}}_1\times\vec{\text{b}}_2\big)$ and $\big|\vec{\text{b}}_1\times\vec{\text{b}}_2\big|$ in equation (1) to get shortest distance between given lines, so
$\text{S.D.}=\Big|\frac{0}{\sqrt{94}}\Big|$
$\text{S.D.}=0$
Since, shortest distance between the given lines is not zero, so lines are intersecting.

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 "5 Marks Questions" 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

Find the equations of the planes parallel to the plane x - 2y + 2z - 3 = 0 and which are at a unit distance from the point (1, 1, 1).
→
Evaluate the following integrals as limit of sum:
$\int\limits^{2}_{0}\big(\text{x}^2+2\big)\text{dx}$
→
Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=(\hat{\text{i}}+\hat{\text{j}})+\lambda(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})+\mu(-\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}})$
→
find the area of the region bounded by y = |x - 1| and y = 1.
→
Let $\text{A}=\begin{bmatrix}-1&1&-1\\3&-3&3\\5&5&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}0&4&3\\1&-3&-3\\-1&4&4\end{bmatrix},$ compute $A^2 - B^2.$
→
Three persons A, B, C throw a die in succession till one gets a 'six' and wins the game. Find their respective probabilities of winning.
→
Solve the following differential equation:
$(\text{x + y})(\text{dx}-\text{dy})=\text{dx + dy}$
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\sin3\text{x}}{\text{x}},&\text{if }\text{ x}\neq0\\4,&\text{if }\text{ x}=0\end{cases}$
→
The mean and variance of the binomial distribution are 4 and $\frac{4}{3}$ respectively. Find the distribution and P(X>1).
→
A hospital dietician wishes to find the cheapest combination of two foods, A and B, that contains at least 0.5 milligram of thiamin and at least 600 calories. Each unit of A contains 0.12 milligram of thiamin and 100 calories, while each unit of B contains 0.10 milligram of thiamin and 150 calories. If each food costs 10 paise per unit, how many units of each should be combined at a minimum cost?
→