« Line in three dimensions 

Plane through $(x_0, y_0, z_0)$ and perpendicular to the direction $(a, b, c)$:
$a (x  x_0) + b (y  y_0) + c (z  z_0) = 0$
Example 1
Find the equation for the plane through the point $(0, 1, 2)$ perpendicular to the vector $(2, 1, 3)$.
Solution:
$(x_0, y_0, z_0) = (0, 1, 2)$
$(a, b, c) = (2, 1, 3)$
The plane: $2 (x  0) + 1 (y  1)  3 (z  2) = 0$
$2x + y 3z = 5$.
Plane through $(x_0, y_0, z_0)$, $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$:
$$ \left {\begin{array}{*{20}{c}} {x  {x_0}}&{y  {y_0}}&{z  {z_0}}\\ {{x_1}  {x_0}}&{{y_1}  {y_0}}&{{z_1}  {z_0}}\\ {{x_2}  {x_0}}&{{y_2}  {y_0}}&{{z_2}  {z_0}} \end{array}} \right = 0 $$
Example 2:
Find the equation for the plane through the points $(0, 1, 2)$, $(2, 1, 3)$ and $(3, 1, 0)$
Solution:
$$ \begin{aligned} &\left {\begin{array}{*{20}{c}} {x  0}&{y  1}&{z  2}\\ {2  0}&{1  1}&{3  2}\\ {3  0}&{1  1}&{0  2} \end{array}} \right = 0 \\ &\left {\begin{array}{*{20}{c}} x&{y  1}&{z  2}\\ 2&0&1\\ 3&0&{  2} \end{array}} \right = 0 \\ &3(y  1) + 4(y  1) = 0 \\ &y = 1 \\ &0x + 1y + 0z = 0 \end{aligned} $$
Plane through $(x_0, y_0, z_0)$ and parallel to the vectors $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$:
$$ \left {\begin{array}{*{20}{c}} {x  {x_0}}&{y  {y_0}}&{z  {z_0}}\\ {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}} \right = 0 $$
Plane through $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$ and parallel to the direction $(a, b, c)$:
$$ \left {\begin{array}{*{20}{c}} {x  {x_0}}&{y  {y_0}}&{z  {z_0}}\\ {{x_1}  {x_0}}&{{y_1}  {y_0}}&{{z_1}  {z_0}}\\ a&b&c \end{array}} \right = 0 $$
The distance from the point $(x_0, y_0, z_0)$ to the plane $ax + by + cz + d = 0$ is
$$ d = \frac{{\left {a{x_0} + b{x_0} + c{z_0}} \right}}{{\sqrt {{a^2} + {b^2} + {c^2}} }} $$
The angle between two planes $a_0 x + b_0 y +c_0 z + d_0 = 0$ and $a_1 x + b_1 y +c_1 z + d_1 = 0$ is
$$ \varphi = \arccos \frac{{a0a1 + b0b1 + c0c1}}{{\sqrt {{a_0}^2 + {b_0}^2 + {c_0}^2} \sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} }} $$
Two planes are parallel if their normal vectors are parallel (constant multiples of one another). It is easy to recognize parallel planes written in the form $ax + by + cz = d$, since a quick comparison of the normal vectors $n=< a, b, c >$ can be made.