Download App

Line and Plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsLine and Plane3 Marks
Question
Show that lines $\bar{r}=(-\hat{i}-3 \hat{j}+4 \hat{k})+\lambda(-10 \hat{i}-\hat{j}+\hat{k})$ and $\bar{r}=(-10 \hat{i}-\hat{j}+\hat{k}) \mu(-\hat{i}-3 \hat{j}+4 \hat{k})$ intersect each other.
Find the position vector of their point of intersection.
✓

Answer

The position vector of a variable point on the line $\bar{r}=(-\hat{i}-3 \hat{j}+4 \hat{k})+\lambda(-10 \hat{i}-\hat{j}+\hat{k})$ is $(-1-10 \lambda) \hat{i}+(-3-\lambda) \hat{j}+(4+\lambda) \hat{k}$
The position vector of a variable point on the line $\bar{r}=(-10 \hat{i}-\hat{j}+\hat{k})+\mu(-\hat{i}-3 \hat{j}+4 \hat{k})$ is $(-10-1 \mu) \hat{i}+(-1-3 \mu) \hat{j}+(1+4 \mu) \hat{k}$ Given lines intersect each other if there exist some values of $\lambda$ and $\mu$ for which
$
\begin{aligned}
& (-1-10 \lambda) \hat{i}+(-3-\lambda) \hat{j}+(4+\lambda) \hat{k}=(-10-1 \mu) \hat{i}+(-1-3 \mu) \hat{j}+(1+4 \mu) \hat{k} \\
\therefore \quad & -1-10 \lambda=-10-1 \mu,-3-\lambda=-1-3 \mu \text { and } 4+\lambda=1+4 \mu \\
\therefore \quad & 10 \lambda-\mu,=9, \lambda-3 \mu=-2 \text { and } \lambda-4 \mu=-3
\end{aligned}
$
Given lines intersect each other if this system is consistent
$\operatorname{As}\left|\begin{array}{ccc}10 & -1 & 9 \\ 1 & -3 & -2 \\ 1 & -4 & -3\end{array}\right|=10(9-8)+1(-3+2)+9(-4+3)=10-1-9=0$
$\therefore$ The system (1) is consistent and lines intersect each other.
Solving any two equations in system (1), we get. $\lambda=1, \mu=1$
Substituting this value of $\lambda$ in $(-1-10 \lambda) \hat{i}+(-3-\lambda) \hat{j}+(4+\lambda) \hat{k}$ we get, $-11 \hat{i}-4 \hat{j}+5 \hat{k}$
$\therefore$ The position vector of their point of intersection is $-11 \hat{i}-4 \hat{j}+5 \hat{k}$.

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 "Solve the Following Question.(3 Marks)" from Line and Plane→Line and Plane — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

$\int \sin (\log x) d x$
→
Evaluate the following:

$\int_0^1 t^5 \sqrt{1-t^2} d t$

→
Diffrentiate the following w. r. t. x.
$\tan ^{-1}(\operatorname{cosec} x+\cot x)$
→
Represent the following families of curves by forming the corresponding differential equation:
$\text{y}=\text{ax}^3$
→
Write the following in the simplest form:
$\sin^{-1}\Big\{\frac{\text{x}+\sqrt{1-\text{x}^2}}{\sqrt{2}}\Big\},-1<\text{x}<1$
→
If $f'(x) = 8x^3 - 2x, f'(2) = 8,$ find $f'(x)$.
→
Differentiate the following w. r. t. x.$\sin ^{-1}\left(\frac{5 \sqrt{1-x^2}-12 x}{13}\right)$
→
Evaluate the following integrals:$\int\frac{\text{x}+5}{3\text{x}^2+13\text{x}-10}\text{ dx}$
→
Find the nth derivative of the following:
log(ax + b)
→
Suppose that $80 \%$ of all families own a television set. If 10 families are interviewed at random, find the probability that at most three families own a television set.
→