The equation of the plane, which bisects the | vedclass

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

Question

(C) Let the points be $A(1, 2, 3)$ and $B(3, 4, 5)$.The midpoint $M$ of the line segment $AB$ is given by $M = \left(\frac{1+3}{2}, \frac{2+4}{2}, \fr

Vedclass · Accepted Answer

$x+y+z=9$

Vedclass · Answer

(C) Let the points be $A(1, 2, 3)$ and $B(3, 4, 5)$.The midpoint $M$ of the line segment $AB$ is given by $M = \left(\frac{1+3}{2}, \frac{2+4}{2}, \frac{3+5}{2}\right) = (2, 3, 4)$.The direction ratios of the line segment $AB$ are $(3-1, 4-2, 5-3) = (2, 2, 2)$.Since the plane bisects $AB$ at right angles, the line $AB$ is normal to the plane. Thus, the normal vector is $\vec{n} = 2\hat{i} + 2\hat{j} + 2\hat{k}$, which can be simplified to $\vec{n} = \hat{i} + \hat{j} + \hat{k}$.The equation of a plane passing through a point $\vec{a}$ with normal vector $\vec{n}$ is given by $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$.Here, $\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ and $\vec{n} = \hat{i} + \hat{j} + \hat{k}$.Substituting these values, we get: $((x\hat{i} + y\hat{j} + z\hat{k}) - (2\hat{i} + 3\hat{j} + 4\hat{k})) \cdot (\hat{i} + \hat{j} + \hat{k}) = 0$.$(x-2)\hat{i} + (y-3)\hat{j} + (z-4)\hat{k} \cdot (\hat{i} + \hat{j} + \hat{k}) = 0$.$(x-2) + (y-3) + (z-4) = 0$.$x + y + z - 9 = 0$, or $x + y + z = 9$.

The equation of the plane, which bisects the line joining the points $(1, 2, 3)$ and $(3, 4, 5)$ at right angles is

Similar Questions

The equation of a plane parallel to the $x$-axis is:

The equation of the plane bisecting the line segment joining the points $(2,0,6)$ and $(-6,2,4)$ and perpendicular to it,is

Let the plane $ax+by+cz+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at right angles. If $a, b, c, d$ are integers,then the minimum value of $(a^2+b^2+c^2+d^2)$ is

Find the vector and Cartesian equations of the plane that passes through the point $(1, 4, 6)$ and the normal vector to the plane is $\hat{i} - 2\hat{j} + \hat{k}$.

The equation $xy = 0$ in three-dimensional space represents

Vedclass Test Series

Exam Paper Generator

Online Exam Module