If L_1 and L_2 are two lines which pass throu | vedclass

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

Question

(C) Since the lines $L_1$ and $L_2$ pass through the origin $(0, 0, 0)$,the plane containing them also passes through the origin. The equation of such

Vedclass · Accepted Answer

$2x - y + z = 0$

Vedclass · Answer

(C) Since the lines $L_1$ and $L_2$ pass through the origin $(0, 0, 0)$,the plane containing them also passes through the origin. The equation of such a plane is given by $ax + by + cz = 0$.The normal vector $\vec{n}$ to the plane is the cross product of the direction vectors of the two lines,$\vec{v}_1 = 3\hat{i} + \hat{j} - 5\hat{k}$ and $\vec{v}_2 = 2\hat{i} + 3\hat{j} - \hat{k}$.$\vec{n} = \vec{v}_1 	imes \vec{v}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & 1 & -5 \ 2 & 3 & -1 \end{vmatrix}$$\vec{n} = \hat{i}(-1 - (-15)) - \hat{j}(-3 - (-10)) + \hat{k}(9 - 2)$$\vec{n} = 14\hat{i} - 7\hat{j} + 7\hat{k}$Dividing by $7$,we get the normal vector as $2\hat{i} - \hat{j} + \hat{k}$.Thus,the equation of the plane is $2(x - 0) - 1(y - 0) + 1(z - 0) = 0$,which simplifies to $2x - y + z = 0$.

If $L_1$ and $L_2$ are two lines which pass through the origin and have direction ratios $(3, 1, -5)$ and $(2, 3, -1)$ respectively,then the equation of the plane containing $L_1$ and $L_2$ is

Similar Questions

$A$ plane meets the coordinate axes at $A, B, C$ such that the centroid of the $\Delta ABC$ is the point $(\alpha, \beta, \gamma)$. Show that the equation of the plane is $\frac{x}{\alpha} + \frac{y}{\beta} + \frac{z}{\gamma} = 3$.

The perpendicular distance from the origin to the plane $x + 2y - 2z + 5 = 0$ equals $.........$ units.

If the foot of the perpendicular from $(0,0,0)$ to the plane is $(1,2,2)$,then the equation of the plane is

Find the Cartesian equation of the following plane: $\vec{r} \cdot (2\hat{i} + 3\hat{j} - 4\hat{k}) = 1$

The equation of the plane,passing through the point $(-1, 2, -3)$ and parallel to the lines $\frac{x-1}{3} = \frac{y-2}{2} = \frac{z}{-4}$ and $\frac{x}{2} = \frac{y-1}{-3} = \frac{z-2}{2}$,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module