Plane passing through a point and perpendicular to a vector (vector form)
Plane Passing Through a Point and Perpendicular to a Vector (Vector Form)
In 3D geometry, a plane can be uniquely determined by a point through which it passes and a vector that is perpendicular to it. This vector is often referred to as the normal vector of the plane. The equation of such a plane can be expressed in vector form, which is particularly useful in various applications such as computer graphics, physics, and engineering.
Understanding the Vector Form of a Plane
The vector form of the equation of a plane is based on the concept of the dot product. The dot product of two vectors is a scalar quantity that is a measure of the vectors' parallelism. It is zero if the vectors are perpendicular.
If a plane passes through a point ( P_0(x_0, y_0, z_0) ) and is perpendicular to a vector ( \vec{n} = (a, b, c) ), then any point ( P(x, y, z) ) on the plane satisfies the following condition:
[ \vec{P_0P} \cdot \vec{n} = 0 ]
where ( \vec{P_0P} ) is the vector from ( P_0 ) to ( P ), given by ( (x - x_0, y - y_0, z - z_0) ).
The Equation of the Plane
Using the dot product, the equation of the plane can be written as:
[ (x - x_0, y - y_0, z - z_0) \cdot (a, b, c) = 0 ]
Expanding the dot product, we get the scalar equation of the plane:
[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 ]
This can be further simplified to:
[ ax + by + cz = d ]
where ( d = ax_0 + by_0 + cz_0 ).
Table of Differences and Important Points
| Feature | Description |
|---|---|
| Point ( P_0 ) | A given point through which the plane passes. It is used to anchor the plane in space. |
| Normal Vector ( \vec{n} ) | A vector that is perpendicular to the plane. It defines the orientation of the plane. |
| Vector ( \vec{P_0P} ) | A position vector from the point ( P_0 ) to any point ( P ) on the plane. |
| Dot Product | A scalar product of two vectors that is zero when the vectors are perpendicular. |
| Equation of Plane | A linear equation in ( x ), ( y ), and ( z ) that represents all points on the plane. |
Formulas
- Vector form of the plane: ( \vec{P_0P} \cdot \vec{n} = 0 )
- Scalar equation of the plane: ( ax + by + cz = d )
Examples
Example 1: Finding the Equation of a Plane
Given a point ( P_0(1, 2, 3) ) and a normal vector ( \vec{n} = (4, -2, 5) ), find the equation of the plane.
Solution:
Using the scalar equation of the plane:
[ d = 4(1) - 2(2) + 5(3) = 4 - 4 + 15 = 15 ]
The equation of the plane is:
[ 4x - 2y + 5z = 15 ]
Example 2: Checking if a Point Lies on the Plane
Given the plane ( 2x + 3y - z = 6 ) and a point ( Q(1, 1, -1) ), determine if ( Q ) lies on the plane.
Solution:
Substitute the coordinates of ( Q ) into the plane's equation:
[ 2(1) + 3(1) - (-1) = 2 + 3 + 1 = 6 ]
Since the left-hand side equals the right-hand side, point ( Q ) lies on the plane.
Understanding the concept of a plane passing through a point and perpendicular to a vector is crucial for solving problems in 3D geometry. By using the vector form of the equation of a plane, one can easily determine the plane's equation, check if a point lies on the plane, and perform other geometric operations.