Name: plane equations from a point and normal
Uploaded: 2026-06-08T09:39:09.242Z
Duration: 1 min 31 s
Channel: JoVE

三维空间中的一个平面由平面上的一个点和一个法向量定义。该向量垂直于平面表面，并确定平面的方向。在实际应用中，这模拟了附着在建筑物上的倾斜玻璃板的表面。面板底面的角位于一个已知点，该点由位置矢量 r0 表示。其方向由一个垂直于表面的法向矢量 n 确定。面板上的第二个点由另一个位置矢量 r 定义。将 r0 从 r 中减去，应用矢量差法则，得到一个位于面板表面的矢量。由于该向量位于平面上，因此它与n正交，且它们的点积等于零。这种正交性构成了平面的向量方程。重写该方程可得到其另一种形式。将点积展开为 x-，y-， 和 z-分量可得到标量方程。该方程利用法向量的分量和固定基点的坐标。该方程定义了位于面板表面上的所有点。将面板建模为平面有助于检查安装点是否满足结构对准方程。

空间中的平面

Planes in Space

三维空间中的一个平面从根本上由位于该平面上的一个点和一个垂直于其表面的法向量所确定。该法向量唯一地决定了平面的方向，使其成为一种基本的几何描述量。在建筑应用中，例如在建筑物立面上安装倾斜的玻璃面板时，这种数学模型能够精确地表示面板在空间中的位置和方向。 设 r₀ 为玻璃板上某一已知点的位置矢量（例如，底座安装点），n 为垂直于板面的法向量。平面上任意其他点 r 满足如下条件：位于平面上的矢量 r - r₀ 与 n 正交。由此可得平面的矢量方程：n · (r - r₀) = 0该表达式描述了法向量与平面内任意向量之间的正交条件。展开点积可得到平面方程的标量形式。若 n = ⟨a, b, c⟩ 且 r₀ = ⟨x₀, y₀, z₀⟩，则标量方程变为：a(x - x₀) + b(y - y₀) + c(z - z₀) = 0。该标量形式描述了位于平面上的所有点（x, y, z），在实际计算中尤为有用。利用该数学模型，工程师可以验证面板的拟定安装点是否与目标表面一致。通过将每个安装点的坐标代入平面方程，可以判断其是否满足共面条件。这能够确保建筑构件（如玻璃面板）的结构完整性和精确安装。

https://cloudfront.jove.com/files/ftp_upload/40633/40633_article_thumb_40633.jpg

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Vectors in Space

空间向量

KEY TERMS AND CONCEPTS

关键术语与概念

太空中的平面

plane equations from a point and normal

A plane in three-dimensional space can be described by one point on the plane and a normal vector. The normal vector is perpendicular to the surface, and it sets the plane’s orientation. In geometry, this gives a clear way to model a flat surface in space.

The transcript uses a sloped glass panel on a building façade as a real-world example. Let r₀ be the position vector of a known point on the panel, such as the base mounting point. Let n be the normal vector to the panel. Any other point r on the plane makes the vector r - r₀ lie in the plane, so that vector must be orthogonal to n.

That idea leads to the vector equation of the plane: n · (r - r₀) = 0. The dot product here shows the orthogonality condition between the normal vector and any vector that lies on the plane. If n = ⟨a, b, c⟩ and r₀ = ⟨x₀, y₀, z₀⟩, the equation can be written in scalar form as a(x - x₀) + b(y - y₀) + c(z - z₀) = 0.

The scalar equation describes every point (x, y, z) that lies on the plane. It is useful for practical calculations because engineers can test proposed mounting points by substituting their coordinates into the equation. If the points satisfy the condition, they are planar and fit the intended surface. This helps confirm the structural integrity and accurate installation of architectural components such as glass panels.

Plane Equations from a Point and Normal

introduction to vectors

Vectors in Three-Dimensional Motion

Vectors describe motion in three-dimensional space with both size and direction. They are useful for quantities such as displacement and velocity, since those measurements need more than just a number. A vector is shown as an arrow from one point to another. The arrow’s length gives the magnitude, and its direction shows the way the motion points. In three dimensions, a vector is written in terms of its x-, y-, and z-components. These components are often shown with angle brackets, such as ⟨x,...

vectors in engineering applications

Cable Tension and Force Balance

Cable tension and force balance are shown with a steel beam held by two identical cables. The beam has a downward weight of 5000 N. The cables support it from opposite sides. Because the setup is symmetric, each cable makes a 60° angle with the horizontal beam and carries the same tension. The beam stays completely at rest, so it is in static equilibrium. Static equilibrium means the total horizontal force and the total vertical force are both zero. Each cable pulls in its own direction, so...

dot product of two vectors

Vector Projection and the Dot Product

The dot product connects vectors, angles, and projections in vector algebra. It combines two vectors to produce a scalar, which is a single number. This makes it useful in math and in physics, especially for work. For vectors a = ⟨a1, a2, a3⟩ and b = ⟨b1, b2, b3⟩, the dot product is found by multiplying matching components and adding them: a · b = a1b1 + a2b2 + a3b3. The operation is commutative, so the order does not matter. It is also distributive over vector addition and works with scalar...

cross product and its geometry

Cross Product, Area, and Torque

The cross product connects vector direction, area, and rotation in three-dimensional space. When two non-zero vectors are not parallel, they define a unique plane and form a parallelogram. Their cross product gives a third vector that is perpendicular to that plane. The direction of this new vector is found with the Right-Hand Rule. Point the fingers of your right hand along the first vector and curl them toward the second vector. Your thumb shows the direction of the result. This convention...

lines in space

Vector Form for a Line in Space

A line in three-dimensional space can be described with a vector equation. This method uses a known point on the line and a direction vector. It is a useful way to model straight paths in space. The transcript uses a surveying laser beam as an example. A laser beam from a surveying instrument can be treated as a line across a construction site. The beam starts at a known point, called P₀, whose position vector is r₀. The beam also has a direction vector v, which is parallel to its path. Any...

planes in space

A plane in three-dimensional space can be described by one point on the plane and a normal vector. The normal vector is perpendicular to the surface, and it sets the plane’s orientation. In geometry, this gives a clear way to model a flat surface in space. The transcript uses a sloped glass panel on a building façade as a real-world example. Let r₀ be the position vector of a known point on the panel, such as the base mounting point. Let n be the normal vector to the panel. Any other point r...

cylinders in three-dimensional space

Modeling Cylindrical Surfaces in 3D

Cylindrical surfaces in three-dimensional space come from moving a two-dimensional curve in a straight line. As the curve slides, it leaves behind translated copies that build a surface. These copies form parallel rulings, which are straight lines in the same fixed direction. This idea helps describe some 3D shapes with simple equations in Cartesian coordinates. A cylindrical surface is often shown by an equation that leaves out one of the three variables. For example, y = x^2 does not include...

quadric surfaces

Shapes of Quadric Surfaces in 3D

Quadric surfaces are smooth three-dimensional shapes described by second-degree equations in x, y, and z. The way the squared and linear terms are arranged determines the surface. The main quadric surfaces are ellipsoids, cones, paraboloids, and hyperboloids. An ellipsoid is a closed surface that appears when all three variables are squared with the same sign. It is bounded and symmetric about its principal axes. Cross-sections taken along those axes produce circles or ellipses. A standard...

vectors in 2d: problem solving

Crosswind Vector Navigation in 2D

Crosswind vector navigation in 2D shows how wind changes an airplane’s path. A plane flew due north at 180 km/h in still air, but after 30 minutes it was 80 km off course. Its track was about 5 degrees east of north, which shows that a crosswind pushed it away from the intended route. The plane’s ground path was diagonal, not straight north. That means its effective ground speed was reduced to 160 km/h and shifted slightly east. By studying the displacement from the planned path, the wind’s...

vectors in space: problem solving

Finding Cable Tensions in 3D Equilibrium

Three-dimensional static equilibrium can be used to find the tension in cables that support a chandelier. In this setup, the chandelier weighs 1000 N and sits at the origin of a 3D coordinate system. Three ceiling anchor points are located above it, and each cable runs from the chandelier to one anchor point. To analyze the forces, the direction of each cable must first be described with a position vector. A position vector is a vector that points from the chandelier to an anchor point. These...

key terms and concepts

Learn how a point and a normal vector define a plane in 3D space, and how to use the plane equation to check mounting points on a glass panel.

Plane Equations from a Point and Normal | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Plane Equations from a Point and Normal

Description

Transcript