The point of intersection of the lines repres | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the point of intersection be $P$. The coordinates of any point on the first line are $(1+2\lambda, 2+3\lambda, -1+4\lambda)$ and on the second

Vedclass · Accepted Answer

$3 \hat{i}+5 \hat{j}+3 \hat{k}$

Vedclass · Answer

(A) Let the point of intersection be $P$. The coordinates of any point on the first line are $(1+2\lambda, 2+3\lambda, -1+4\lambda)$ and on the second line are $(-1+\mu, -3+2\mu, 7-\mu)$.Equating the coordinates,we get:$1+2\lambda = -1+\mu \implies 2\lambda - \mu = -2$ ... $(i)$$2+3\lambda = -3+2\mu \implies 3\lambda - 2\mu = -5$ ... $(ii)$$-1+4\lambda = 7-\mu \implies 4\lambda + \mu = 8$ ... $(iii)$Adding equations $(i)$ and $(iii)$:$(2\lambda - \mu) + (4\lambda + \mu) = -2 + 8$$6\lambda = 6 \implies \lambda = 1$Substituting $\lambda = 1$ in equation $(i)$:$2(1) - \mu = -2 \implies \mu = 4$Checking these values in equation $(ii)$:$3(1) - 2(4) = 3 - 8 = -5$,which is correct.Substituting $\lambda = 1$ in the first line equation:$r = (\hat{i}+2 \hat{j}-\hat{k}) + 1(2 \hat{i}+3 \hat{j}+4 \hat{k}) = 3 \hat{i}+5 \hat{j}+3 \hat{k}$.Thus,the point of intersection is $3 \hat{i}+5 \hat{j}+3 \hat{k}$.

The point of intersection of the lines represented by $r=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $r=(-\hat{i}-3 \hat{j}+7 \hat{k})+\mu(\hat{i}+2 \hat{j}-\hat{k})$ is

Similar Questions

Point $A$ lies at a distance of $6$ units from the point $(1, 0, 1)$ on the line $\frac{x - 1}{2} = \frac{y}{2} = \frac{z - 1}{1}$ in the negative $z$-direction. Find the coordinates of $A$.

The Cartesian equation of the line which is parallel to the vector $3 \hat{i} + 2 \hat{j} - 8 \hat{k}$ and passes through the point $(5, 2, -4)$ is . . . . . . .

The lines $\overrightarrow{r} = (\hat{i} - \hat{j}) + \ell(2\hat{i} + \hat{k})$ and $\overrightarrow{r} = (2\hat{i} - \hat{j}) + m(\hat{i} + \hat{j} - \hat{k})$:

If the shortest distance between the lines $\vec{r}_{1}=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0$ and $\vec{r}_{2}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k}), \mu \in R$ is $9$,then $\alpha$ is equal to $.....$

If the $x$-coordinate of a point $P$ on the line joining the points $Q(2, 2, 1)$ and $R(5, 2, -2)$ is $4$,then the $y$-coordinate of $P$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module