evinda said:The line equation that passes through $a$ and is parallel to $b$ is $l(t)=a+tb$.
From the equation of the plane $x + 2y + 3z = 12$, we have that a normal vector to the plane is $(1,2,3)$.
We are looking for a line that is normal to the plane, so parallel to the vector $(1,2,3)$ and passes through the point $(4,6,8)$.
Can you find now the line equation?
A normal vector is a vector that is perpendicular, or at a 90 degree angle, to a given surface or plane. In other words, it is a vector that is perpendicular to all vectors in a given plane.
To find the equation of a plane with a given normal vector and point, you can use the formula Ax + By + Cz = D, where A, B, and C are the components of the normal vector and x, y, and z are the coordinates of the point. The value of D can be found by plugging in the values of x, y, and z from the given point into the equation.
Finding the equation of a plane allows us to understand the relationship between different points and vectors in 3D space. It also helps us to determine the orientation of the plane and its distance from the origin.
No, to find the equation of a plane, you need at least three non-collinear points or a normal vector and a point. Having only two points is not enough information to determine a unique plane.
Yes, there are infinite equations that can represent a given plane. This is because planes are infinite in all directions and can be translated and rotated without changing the fundamental properties of the plane.