Example. Write the equation of the plane passing through the point $M_{0}(2,5,-3)$ and perpendicular to the vector $\overline{M_{1} M_{2}}$, where points $M_{1}$ and $M_{2}$ have coordinates $(7,8,-1)$ and $(9,7,4)$.
Solution.

1. As the normal vector of the plane $\vec{n}$, we choose the vector $\overline{M_{1} M_{2}}=\{2,-1,5\}$.
2. We form the equation of the plane (1) with the normal vector $\vec{n}=\{2,-1,5\}$, passing through the point $M_{0}(2,5,-3)$:

$$
2(x-2)-1(y-5)+5(z+3)=0
$$

Answer. The equation of the plane is $2 x-y+5 z+16=0$.

Problem Conditions. Write the equation of the plane passing through the point $M_{0}$ and perpendicular to the vector $\overline{M_{1} M_{2}}$.

1. $M_{0}(3,2,0)$

$M_{1}(4,1,5)$

$M_{2}(2,-1,4)$.
2. $M_{0}(-5,-1,0)$

$M_{1}(-5,1,-4)$

$M_{2}(-2,2,-3)$.
3. $M_{0}(2,-4,-2)$,

$M_{1}(-1,-3,-7)$

$M_{2}(-4,-1,-5)$.
4. $M_{0}(-5,3,10)$,

$M_{1}(0,5,7)$,

$M_{2}(2,7,8)$.
5. $M_{0}(2,-10,-4), \quad M_{1}(0,-6,-8), \quad M_{2}(-2,-5,-9)$.
6. $M_{0}(1,9,2)$

$M_{1}(0,4,7)$

$M_{2}(1,6,9)$.
7. $M_{0}(0,-2,7)$,

$M_{1}(-5,-4,9)$,

$M_{2}(-2,-2,6)$.
8. $M_{0}(-1,1,-4)$,

$M_{1}(3,8,-2)$

$M_{2}(2,11,0)$.
9. $M_{0}(-1,7,-6)$,

$M_{1}(3,5,-1)$

$M_{2}(1,3,-2)$.
10. $M_{0}(-5,2,5)$

$M_{1}(3,-3,-2)$,

$M_{2}(4,-1,2)$.

Answers.

1. $2 x+2 y+z=0$.
2. $3 x+y+z-4=0$.
3. $3 x-2 y-2 z-16=0$.
4. $2 x+2 y+z+9=0$.
5. $2 x-y+z-20=0$.
6. $x+2 y+2 z-3=0$.
7. $3 x+2 y-3 z+23=0$.
8. $x-3 y-2 z-8=0$.
9. $2 x+2 y+z-16=0$.
10. $x+2 y+4 z-5=0$.

## 1.9. Angle Between Planes

Problem Statement. Find the angle between the planes

$$
A_{1} x+B_{1} y+C_{1} z+D_{1}=\cdot 0, \quad A_{2} x+B_{2} y+C_{2} z+D_{2}=0
$$

Plan of Solution. The dihedral angle between the planes is equal to the angle between their normal vectors

$$
\vec{n}_{1}=\left\{A_{1}, B_{1}, C_{1}\right\}, \quad \vec{n}_{2}=\left\{A_{2}, B_{2}, C_{2}\right\}
$$

Therefore, the angle $\varphi$ between the planes is determined by the equality

$$
\cos \varphi=\frac{\left(\vec{n}_{1}, \vec{n}_{2}\right)}{\left|\vec{n}_{1}\right| \cdot\left|\vec{n}_{2}\right|}
$$