Plane passing through two points and parallel to a vector (vector form)
Plane Passing Through Two Points and Parallel to a Vector (Vector Form)
In 3D geometry, a plane can be uniquely determined by a point and a normal vector to the plane. However, there are situations where we need to determine a plane that passes through two given points and is parallel to a given vector. This scenario can be approached using vector algebra.
Understanding the Concept
A plane in 3D space can be defined using a point and a normal vector. However, when we want a plane to pass through two specific points and be parallel to a given vector, we need to find a normal vector that is perpendicular to the given vector and also contains the two points.
Vector Form of a Plane
The vector equation of a plane is given by:
[ \vec{r} \cdot \vec{n} = d ]
where:
- ( \vec{r} ) is the position vector of any point on the plane,
- ( \vec{n} ) is the normal vector to the plane,
- ( d ) is the perpendicular distance from the origin to the plane.
Finding the Normal Vector
To find the normal vector ( \vec{n} ), we need to use the cross product of two non-parallel vectors that lie on the plane. If we have two points ( A ) and ( B ) and a vector ( \vec{v} ) that the plane is parallel to, we can find a vector ( \vec{AB} ) that lies on the plane by subtracting the position vectors of ( B ) from ( A ). The normal vector ( \vec{n} ) can then be found by taking the cross product of ( \vec{AB} ) and ( \vec{v} ):
[ \vec{n} = \vec{AB} \times \vec{v} ]
Formulating the Plane Equation
Once we have the normal vector ( \vec{n} ), we can use the position vector of either point ( A ) or ( B ) to formulate the plane equation. Let's use point ( A ) with position vector ( \vec{a} ):
[ \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} ]
This equation represents the plane that passes through point ( A ) and is perpendicular to ( \vec{n} ).
Differences and Important Points
| Aspect | Plane through a Point and Normal Vector | Plane through Two Points and Parallel to a Vector |
|---|---|---|
| Definition | Defined by a point and a normal vector. | Defined by two points and a parallel vector. |
| Normal Vector | Given directly or found using cross product. | Found using cross product of the vector between two points and the parallel vector. |
| Equation Form | ( \vec{r} \cdot \vec{n} = d ) | ( \vec{r} \cdot (\vec{AB} \times \vec{v}) = \vec{a} \cdot (\vec{AB} \times \vec{v}) ) |
| Determination of ( d ) | Given or calculated using a known point. | Calculated using the position vector of one of the points. |
Examples
Example 1: Finding the Plane Equation
Suppose we have two points ( A(1, 2, 3) ) and ( B(4, 5, 6) ), and the plane is parallel to the vector ( \vec{v} = \begin{bmatrix} 7 \ 8 \ 9 \end{bmatrix} ).
- Find the vector ( \vec{AB} ):
[ \vec{AB} = \begin{bmatrix} 4 - 1 \ 5 - 2 \ 6 - 3 \end{bmatrix} = \begin{bmatrix} 3 \ 3 \ 3 \end{bmatrix} ]
- Calculate the normal vector ( \vec{n} ):
[ \vec{n} = \vec{AB} \times \vec{v} = \begin{bmatrix} 3 \ 3 \ 3 \end{bmatrix} \times \begin{bmatrix} 7 \ 8 \ 9 \end{bmatrix} = \begin{bmatrix} (3 \cdot 9 - 3 \cdot 8) \ (3 \cdot 7 - 3 \cdot 9) \ (3 \cdot 8 - 3 \cdot 7) \end{bmatrix} = \begin{bmatrix} 3 \ -6 \ 3 \end{bmatrix} ]
- Formulate the plane equation using point ( A ):
[ \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} ]
[ \vec{r} \cdot \begin{bmatrix} 3 \ -6 \ 3 \end{bmatrix} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} \cdot \begin{bmatrix} 3 \ -6 \ 3 \end{bmatrix} ]
[ 3x - 6y + 3z = 3 - 12 + 9 ]
[ 3x - 6y + 3z = 0 ]
This is the equation of the plane passing through points ( A ) and ( B ) and parallel to vector ( \vec{v} ).
Example 2: Verifying a Point Lies on the Plane
Let's verify if point ( C(7, 8, 9) ) lies on the plane found in Example 1.
- Use the plane equation ( 3x - 6y + 3z = 0 ) and substitute the coordinates of point ( C ):
[ 3(7) - 6(8) + 3(9) = 21 - 48 + 27 = 0 ]
Since the equation holds true for point ( C ), it lies on the plane.
By understanding these concepts and following the steps outlined, one can determine the equation of a plane passing through two points and parallel to a given vector, which is a common problem in 3D geometry and can be essential for exams and practical applications in fields such as physics, engineering, and computer graphics.