Vector Equations of a Plane
Parametric Form: r = a + λb + μc
The vector equation of a plane in its parametric form is given by:
$r = a + λb + μc$
Where:
- $r$ is the position vector of any point on the plane
- $a$ is the position vector of a fixed point on the plane
- $b$ and $c$ are non-parallel vectors lying within the plane
- $λ$ and $μ$ are scalar parameters
This equation represents all points on the plane as a linear combination of two non-parallel vectors ($b$ and $c$) added to a fixed point ($a$).Note
The vectors $b$ and $c$ must be non-parallel to ensure they span the entire plane.
Interpretation
- The vector $a$ anchors the plane at a specific point in space.
- Vectors $b$ and $c$ define the "directions" in which the plane extends.
- By varying $λ$ and $μ$, we can reach any point on the plane.
Let \(\mathbf{a} = (1,2,3) \), \(\mathbf{b} = (1,0,1) \), and \(\mathbf{c} = (0,1,-1) \). The equation of the plane is: \[\mathbf{r} = (1,2,3) + \lambda (1,0,1) + \mu (0,1,-1) \] This can be written in component form as: \[\begin{cases} x = 1 + \lambda\\ y = 2 + \mu\\ z = 3 + \lambda - \mu\end{cases} \]
HintUnderstanding Vector Equations: Lines vs. Planes
- A vector line equation extends a point one-dimensionally using a single direction vector, allowing movement along that direction.
- In contrast, a vector plane equation extends a point two-dimensionally using two direction vectors, defining a surface rather than a single path.
Normal Form: (r-a) • n = 0
Another way to represent a plane is using its normal vector:
$(r-a) • n = 0$
This can be restructured to be:
$r • n = a • n$
Where:
- $r$ is the position vector of any point on the plane
- $n$ is a vector normal (perpendicular) to the plane
- $a$ is the position vector of any point on the plane
This equation states that the dot product of the normal vector with any vector from $a$ to a point on the plane ($r - a$) is zero, which is the definition of perpendicularity.Tip
To find a normal vector to a plane given in the form $r = a + λb + μc$, you can use the cross product: $n = b × c$
