Download App

Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space5 Marks
Question
Prove that the line $\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}-\hat{\text{j}}\big)$ and $\vec{\text{r}}=\big(4\hat{\text{i}}-\hat{\text{k}}\big)+\mu\big(2\hat{\text{i}}+3\hat{\text{k}}\big)$ intersect and find their point of intersection.
✓

Answer

The position vectors of two arbitrary points on the given lines are
$\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}-\hat{\text{j}}\big)=(1+3\lambda)\hat{\text{i}}+(1-\lambda)\hat{\text{j}}-\hat{\text{k}}$
$\big(4\hat{\text{i}}-\hat{\text{k}}\big)+\mu\big(2\hat{\text{i}}+3\hat{\text{k}}\big)=(4+2\mu)\hat{\text{i}}+0\hat{\text{j}}+(3\mu-1)\hat{\text{k}}$
If the lines intersect, then they have a common point. so, for some values of $\lambda$ and $\mu,$ we must have
$(1+3\lambda)\hat{\text{i}}+(1-\lambda)\hat{\text{j}}-\hat{\text{k}}=(4+2\mu)\hat{\text{i}}+0\hat{\text{j}}+(3\mu-1)\hat{\text{k}}$
Equating the coefficients of $\hat{\text{i}},\hat{\text{j}}$ and $\hat{\text{k}},$ we get
$1+3\lambda=4+2\mu\dots(1)$
$1-\lambda=0\dots(2)$
$3\mu-1=-1\dots(3)$
Solving (2) and (3), we get
$\lambda=1$
$\mu=0$
Substituting the valuse $\lambda=1$ and $\mu=0$ in (1), we get
$\text{LHS}=1+3\lambda$
$=1+3(1)$
$=4$
$\text{RHS}=4+2\mu$
$=4+2(0)$
$=4$
$\Rightarrow\text{LHS}=\text{RHS}$
Since $\lambda=1$ and $\mu=0$ satisfy (3), the given lines intersect.
Substituting $\mu=0$ in the second line, we get $\vec{\text{r}}=4\hat{\text{i}}+0\hat{\text{j}}-\hat{\text{k}}$ as the position vector of the point of intersection.
Thus, the coordinates of the point of intersection are (4, 0, -1).

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 Straight line in space→Straight line in space — 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 inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 3 & -1 \\ 1 & -1 & 3 \end{bmatrix}$
→
A manufacturer of patent medicines is preparing a production plan on medicines, A and B. There are sufficient raw materials available to make 20000 bottles of A and 40000 bottles of B, but there are only 45000 bottles into which either of the medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes 1 hour to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit is Rs. 8 per bottle for A and Rs. 7 per bottle for B. How should the manufacturer schedule his production in order to maximize his profit?
→
Integrate the function in Exercise:
$\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}$
$\Bigg[\text{Hint:}\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}=\frac{1}{\text{x}^{\frac{1}{3}}\bigg(1+\text{x}^{\frac{1}{6}}\bigg)},\text{put}\ \text{x}=\text{t}^{6}\Bigg]$
→
A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4500 and on the portable model is Rs. 5000. Make an LPP and solve it graphically.
→
Two numbers are selected at random from integers 1 through 9. If the sum is even, find the probability that both the numbers are odd.
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\text{x}^4-16}{\text{x}-2},&\text{if }\text{ x}\neq2\\16,&\text{if }\text{ x}=2\end{cases}$
→
Solve $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}(\log{\text{y}-\log\text{x}+1}).$
→
Let, X denote the number of colleges where you will apply after your results and P (X = x) denotes your probability of getting admission in x number of colleges. It is given that
$ \text{P(x = }x) = \begin{cases} \text{K}x\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }, & \text{if } x = 0 \text{ or 1}\\ 2\text{k}x\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }, & \text{if } x = 2\\ \text{k} (5 - x)\text{ }\text{ }\text{ }\text{ }, &\text{if } x = 3 \text{ or 4'}\\ 0\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }, & \text{if } x > 4 \end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.
→
A publisher sells a hard cover edition of a text book for Rs. $72.00$ and paperback edition of the same ext for Rs. $40.00$. Costs to the publisher are Rs. $56.00$ and Rs. $28.00$ per book respectively in addition to weekly costs of Rs. $9600.00.$ Both types require $5$ minutes of printing time, although hardcover requires $10$ minutes binding time and the paperback requires only $2$ minutes. Both the printing and binding operations have $4,800$ minutes available each week. How many of each type of book should be produced in order to maximize profit?
→
In each of the show that the given differential equation is homogeneous and solve each of them. $\text{x}^2\ \frac{\text{dy}}{\text{dx}}=\text{x}^2-2\text{y}^2+\text{xy}$
→