A straight line is given by vec{r} = (1 + t)h | vedclass

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

Question

(A) The given equation of the line is $\vec{r} = (\hat{i} + \hat{k}) + t(\hat{i} + 3\hat{j} - \hat{k})$,where $t \in R$.Since the line lies in the pla

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(A) The given equation of the line is $\vec{r} = (\hat{i} + \hat{k}) + t(\hat{i} + 3\hat{j} - \hat{k})$,where $t \in R$.Since the line lies in the plane $x + y + cz = d$,every point on the line must satisfy the plane equation.The point $(1, 0, 1)$ lies on the line (at $t = 0$). Substituting this into the plane equation:$1 + 0 + c(1) = d \Rightarrow 1 + c = d$  .....$(1)$Also,the direction vector of the line $\vec{v} = \hat{i} + 3\hat{j} - \hat{k}$ must be perpendicular to the normal vector of the plane $\vec{n} = \hat{i} + \hat{j} + c\hat{k}$.Therefore,their dot product must be zero:$(1)(1) + (3)(1) + (-1)(c) = 0$$1 + 3 - c = 0 \Rightarrow c = 4$.Substituting $c = 4$ into equation $(1)$:$1 + 4 = d \Rightarrow d = 5$.Thus,the value of $(c + d) = 4 + 5 = 9$.

$A$ straight line is given by $\vec{r} = (1 + t)\hat{i} + 3t\hat{j} + (1 - t)\hat{k}$ where $t \in R$. If this line lies in the plane $x + y + cz = d$,then the value of $(c + d)$ is

Similar Questions

The angle between the line $\frac{x + 1}{3} = \frac{y - 1}{4} = \frac{z - 2}{2}$ and the plane $2x - 3y + z + 4 = 0$ is:

What is the distance between the line $r = (2i - 2j + 3k) + \lambda (i - j + 4k)$ and the plane $r \cdot (i + 5j + k) = 5$?

If the plane $4x + 4y - kz = 0$ is the equation of the plane passing through the origin and containing the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$,find the value of $k$.

If the angle between the line $x = \frac{y-1}{2} = \frac{z-3}{\lambda}$ and the plane $x + 2y + 3z = 4$ is $\cos^{-1} \sqrt{\frac{5}{14}}$,then the value of $\lambda$ is

The equation of the plane passing through the line of intersection of the planes $x+y+z=1$ and $3x+4y+5z=2$ and perpendicular to the $XY$-plane is

Vedclass Test Series

Exam Paper Generator

Online Exam Module