Plane Passing Through Two Points and Parallel to a Vector (Cartesian Form)

In 3D geometry, a plane can be defined in several ways. One such method is by specifying two points through which the plane passes and a direction vector to which the plane is parallel. This is particularly useful in applications such as computer graphics, engineering, and physics.

Understanding the Plane Equation

The general equation of a plane in Cartesian form is:

$$ Ax + By + Cz + D = 0 $$

Where (A), (B), (C) are the components of the normal vector to the plane, and (D) is a constant.

Defining a Plane with Two Points and a Vector

To define a plane that passes through two points (P_1(x_1, y_1, z_1)) and (P_2(x_2, y_2, z_2)), and is parallel to a vector (\vec{v} = \langle a, b, c \rangle), we can use the following approach:

1. Find the direction vector (\vec{P_1P_2}) between the two points.
2. Ensure that the direction vector (\vec{v}) is not parallel to (\vec{P_1P_2}) to avoid ambiguity.
3. The cross product of (\vec{P_1P_2}) and (\vec{v}) gives the normal vector (\vec{n}) to the plane.
4. Use point (P_1) (or (P_2)) and the normal vector (\vec{n}) to find the equation of the plane.

Step-by-Step Process

1. Direction Vector (\vec{P_1P_2}): $$ \vec{P_1P_2} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle $$

2. Cross Product (\vec{n}): $$ \vec{n} = \vec{P_1P_2} \times \vec{v} $$

3. Plane Equation: Using point (P_1), the equation of the plane is: $$ n_x(x - x_1) + n_y(y - y_1) + n_z(z - z_1) = 0 $$ where (\vec{n} = \langle n_x, n_y, n_z \rangle).

Differences and Important Points

Aspect	Description
Points on Plane	The plane must pass through the given points (P_1) and (P_2).
Parallel Vector	The plane must be parallel to the given vector (\vec{v}).
Normal Vector	The normal vector (\vec{n}) is perpendicular to the plane.
Cross Product	The cross product of two non-parallel vectors gives a vector perpendicular to both.
Ambiguity	If (\vec{P_1P_2}) is parallel to (\vec{v}), the plane is not uniquely defined.

Example

Let's find the equation of a plane passing through points (P_1(1, 2, 3)) and (P_2(4, 5, 6)), and parallel to the vector (\vec{v} = \langle 7, 8, 9 \rangle).

1. Direction Vector (\vec{P_1P_2}): $$ \vec{P_1P_2} = \langle 4 - 1, 5 - 2, 6 - 3 \rangle = \langle 3, 3, 3 \rangle $$

2. Cross Product (\vec{n}): $$ \vec{n} = \vec{P_1P_2} \times \vec{v} = \langle 3, 3, 3 \rangle \times \langle 7, 8, 9 \rangle $$ $$ \vec{n} = \langle (3 \cdot 9 - 3 \cdot 8), (3 \cdot 7 - 3 \cdot 9), (3 \cdot 8 - 3 \cdot 7) \rangle = \langle -3, -6, 3 \rangle $$

3. Plane Equation: Using point (P_1(1, 2, 3)), the equation of the plane is: $$ -3(x - 1) - 6(y - 2) + 3(z - 3) = 0 $$ Simplifying, we get: $$ -3x + 3 - 6y + 12 + 3z - 9 = 0 $$ $$ -3x - 6y + 3z + 6 = 0 $$ $$ x + 2y - z - 2 = 0 $$

The equation of the plane is (x + 2y - z - 2 = 0).

In conclusion, by following these steps, we can find the equation of a plane given two points and a vector parallel to the plane. This method is essential for solving problems in 3D space and has numerous applications across different fields.