Find the vector and Cartesian equations of th | vedclass

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

Question

(A) The position vector of the point $(1, 0, -2)$ is $\vec{a} = \hat{i} - 2\hat{k}$.The normal vector $\vec{N}$ perpendicular to the plane is $\vec{N}

Vedclass · Accepted Answer

$x + y - z = 3$

Vedclass · Answer

(A) The position vector of the point $(1, 0, -2)$ is $\vec{a} = \hat{i} - 2\hat{k}$.The normal vector $\vec{N}$ perpendicular to the plane is $\vec{N} = \hat{i} + \hat{j} - \hat{k}$.The vector equation of the plane is given by $(\vec{r} - \vec{a}) \cdot \vec{N} = 0$.Substituting the values,we get $(\vec{r} - (\hat{i} - 2\hat{k})) \cdot (\hat{i} + \hat{j} - \hat{k}) = 0$.Let $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ be the position vector of any point $(x, y, z)$ on the plane.Then,$((x - 1)\hat{i} + y\hat{j} + (z + 2)\hat{k}) \cdot (\hat{i} + \hat{j} - \hat{k}) = 0$.Calculating the dot product: $(x - 1)(1) + (y)(1) + (z + 2)(-1) = 0$.$x - 1 + y - z - 2 = 0$.$x + y - z - 3 = 0$.$x + y - z = 3$.Thus,the Cartesian equation of the plane is $x + y - z = 3$.

Find the vector and Cartesian equations of the plane that passes through the point $(1, 0, -2)$ and the normal to the plane is $\hat{i} + \hat{j} - \hat{k}$.

Similar Questions

$A$ vector $\vec{n}$ is inclined to the $x$-axis at $45^\circ$,to the $y$-axis at $60^\circ$,and at an acute angle to the $z$-axis. If $\vec{n}$ is a normal to a plane passing through the point $(\sqrt{2}, -1, 1)$,then the equation of the plane is:

Find the angle between the planes $x+2y+2z-5=0$ and $3x+3y+2z-8=0$.

If the planes $x + 2y + kz = 0$ and $2x + y - 2z = 0$ are at right angles,then the value of $k$ is

The coordinates of the foot of the perpendicular drawn from the origin to the plane $2x - 3y + 4z = 29$ are

$A$ plane $\pi$ makes intercepts $3$ and $4$ respectively on $Z$-axis and $X$-axis. If $\pi$ is parallel to $Y$-axis, then its equation is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module