The equation of the line,passing through (1, | vedclass

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

Question

(A) The line passes through the point $(1, 2, 3)$ and is parallel to the planes $x - y + 2z = 5$ and $3x + y + z = 6$. Let the normal vectors to the p

Vedclass · Accepted Answer

$\frac{x-1}{-3} = \frac{y-2}{5} = \frac{z-3}{4}$

Vedclass · Answer

(A) The line passes through the point $(1, 2, 3)$ and is parallel to the planes $x - y + 2z = 5$ and $3x + y + z = 6$. Let the normal vectors to the planes be $\vec{n_1} = \hat{i} - \hat{j} + 2\hat{k}$ and $\vec{n_2} = 3\hat{i} + \hat{j} + \hat{k}$. The direction vector $\vec{v}$ of the line is parallel to the cross product of the normals: $\vec{v} = \vec{n_1} 	imes \vec{n_2}$. $\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & -1 & 2 \ 3 & 1 & 1 \end{vmatrix} = \hat{i}(-1 - 2) - \hat{j}(1 - 6) + \hat{k}(1 + 3) = -3\hat{i} + 5\hat{j} + 4\hat{k}$. The equation of the line passing through $(x_1, y_1, z_1)$ with direction vector $(a, b, c)$ is $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$. Substituting the values,we get $\frac{x-1}{-3} = \frac{y-2}{5} = \frac{z-3}{4}$.

The equation of the line,passing through $(1, 2, 3)$ and parallel to the planes $x - y + 2z = 5$ and $3x + y + z = 6$,is

Similar Questions

The equation of the plane passing through the intersection of the planes $x+y+z=1$ and $2x+3y-z+4=0$ and parallel to the $X$-axis is

The distance of the point $P(3, 8, 2)$ from the line $\frac{x-1}{2} = \frac{y-3}{4} = \frac{z-2}{3}$ measured parallel to the plane $3x + 2y - 2z + 15 = 0$ is (in $\text{ units}$)

Under what condition is the straight line $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$ parallel to the $xy$-plane?

Find the distance of the point $-\hat{i} - 5\hat{j} - 10\hat{k}$ from the point of intersection of the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}$ and the plane $x - y + z = 5$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module