Question

Question #6f0c5

Answer 1

See the explanation.

Explanation:

The equation of a plane is in the form
`ax+by+cz+d=0`.

In the equation, the vector `vec n=(a,b,c)` is perpendicular to the plane. It is called a normal vector.

However, there is no single equation `f(x,y,z)=0` to express a line in three dimensions.

You have two options to express a 3-D line.
(A) Let `A(x_0,y_0,z_0)` is a certain point on the line, and `vec d=(p,q,r)` is the direction vector, which is parallel to the line.

Then, the equation of the line using parameter `t` is
`(x,y,z)=(x_0+pt, y_0+qt, z_0+rt)`.
This is equivalent to the vector form `vec p=vec a + t vec d`.

(B) Or, you can express the line as an intersection of two planes.

If you solve two equations of planes (of cource, they must not be parallel) as simultaneous equations, the root corresponds to the line.