Question

The equation of plane parallel to xz- plane and passing through 2,-4,0​

Answer 1

Our plane is parallel to XZ plane, so the vector $\vec{a}=\left<0,\ 1,\ 0\right>,$ which is parallel to y axis, is normal to our plane.

Given that the point (2, -4, 0) is a point on our plane. Let the point (x, y, z) be on our plane so that the vector $\vec{b}=\left<x-2,\ y+4,\ z\right>$ lies on our plane.

Now the two vectors $\vec{a}$ and $\vec{b}$ are perpendicular to each other, so their dot product equals zero, i.e.,

$\longrightarrow\vec{a}\cdot\vec{b}=0$

$\longrightarrow\left<0,\ 1,\ 0\right>\cdot\left<x-2,\ y+4,\ z\right>=0$

$\longrightarrow\underline{\underline{y+4=0}}$

This is the equation of our plane.

Another Method:-

If the plane is parallel to XZ plane, then surely y axis is normal to the plane, so every points having the same y coordinate lies on the plane.

Given that the point (2, -4, 0) is a point on the plane. So every points having y coordinate -4 lies on the plane.

Hence the equation of the plane will be,

$\longrightarrow\underline{\underline{y=-4}}$

or,

$\longrightarrow\underline{\underline{y+4=0}}$

Answer 2

$\huge\bf\fbox\red{Answer:-}$

Our plane is parallel to XZ plane, so the vector $\vec{a}=\left < 0,\ 1,\ 0\right > ,$ which is parallel to y axis, is normal to our plane.

Given that the point (2, -4, 0) is a point on our plane. Let the point (x, y, z) be on our plane so that the vector $\vec{b}=\left < x-2,\ y+4,\ z\right >$ lies on our plane.

Now the two vectors $\vec{a}$ and $\vec{b}$ are perpendicular to each other, so their dot product equals zero, i.e.,

$\longrightarrow\vec{a}\cdot\vec{b}=0$

$\longrightarrow\left < 0,\ 1,\ 0\right > \cdot\left < x-2,\ y+4,\ z\right > =0$

$\longrightarrow\underline{\underline{y+4=0}}$

This is the equation of our plane.

Another Method:-

If the plane is parallel to XZ plane, then surely y axis is normal to the plane, so every points having the same y coordinate lies on the plane.

Given that the point (2, -4, 0) is a point on the plane. So every points having y coordinate -4 lies on the plane.

Hence the equation of the plane will be,

$\longrightarrow\underline{\underline{y=-4}}$

or,

$\longrightarrow\underline{\underline{y+4=0}}$.