Name: vector projection and the dot product
Uploaded: 2026-06-08T09:39:04.262Z
Duration: 1 min 27 s
Channel: JoVE

点积将两个向量结合，生成一个标量值。从代数角度，点积通过将向量的对应分量相乘并累加其乘积来计算。从几何角度，它等于两个向量的模长乘积与它们之间夹角余弦值的乘积。一个实际应用是优化太阳能电池板的倾斜角度，以实现最大能量吸收。阳光被建模为一个三维方向向量。另一个矢量表示面板的法线方向，垂直于其表面。由于阳光朝向面板传播，其矢量方向指向面板内部。为了与向外的法线方向对齐，需将阳光矢量反向，然后计算面板法线与反向后的阳光矢量之间的点积。这意味着将对应的分量相乘并相加结果。将该乘积除以向量的模长，可得到余弦项，该余弦项通过测量阳光照射到面板上的直接程度来控制能量吸收。当角度较小时，余弦值较大，意味着能量吸收更多。当角度为90度时，点积为零，面板无法接收到任何直接的太阳能。

两个向量的点积

Dot Product of Two Vectors

点积，或称标量积，是向量代数中一种基本运算，它将两个向量结合生成一个标量。该运算在物理应用中尤为重要，例如用于计算功；在数学背景下，也常用于确定向量投影和方向余弦。对于向量 a = ⟨a₁, a₂, a₃⟩ 和 b = ⟨b₁, b₂, b₃⟩，点积定义为：\begin{equation*}\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\end{equation*}该运算对向量加法满足交换律和分配律，且与标量乘法相容。如果向量 a 和 b 相互正交，则它们的点积等于零。当向量方向相同时，其点积在模长上达到最大值。点积还有一个涉及两个向量之间夹角 θ 的几何定义：\begin{equation*}\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta\end{equation*}该公式用于确定向量之间的夹角，或检验向量是否正交（θ = 90° 意味着 a·b = 0）。向量与正坐标轴之间的夹角称为方向角。如果 α、β 和 γ 分别是向量 a 与 x 轴、y 轴和 z 轴之间的夹角，则方向余弦为：\begin{equation*}cos α = a₁/|𝐚|， cos β = a₂/|𝐚|， cos γ = a₃/|𝐚|\end{equation*}这些满足以下恒等式：\begin{equation*}\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\end{equation*}向量 b 在 a 上的标量投影（compₐ b）和向量投影（projₐ b）通过将 a 方向上的单位向量进行缩放，从而扩展了 compₐ b：\begin{align*}\text{comp}_{\mathbf{a}} \mathbf{b} &amp;= \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\\\text{proj}_{\mathbf{a}} \mathbf{b}&amp; = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2} \mathbf{a}\end{align*}这些投影描述了 b 在 a 方向上的分量大小，在力学和电磁学中具有重要的物理意义。

https://cloudfront.jove.com/files/ftp_upload/40630/40630_article_thumb_40630.png

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

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 multiplication.

The dot product also has a geometric form. If θ is the angle between two vectors, then a · b = |a||b| cos θ. This form helps find the angle between vectors and check whether they are orthogonal, which means at right angles. When θ = 90°, the dot product is zero. When two vectors point in the same direction, the dot product is largest for their magnitudes.

Direction angles describe the angles between a vector and the positive x-, y-, and z-axes. If α, β, and γ are those angles for vector a, then the direction cosines are cos α = a1/|a|, cos β = a2/|a|, and cos γ = a3/|a|. These values satisfy cos²α + cos²β + cos²γ = 1.

The dot product also leads to projection. The scalar projection, or compa b, tells how much of b lies along a, and it is given by compa b = (a · b)/|a|. The vector projection, or proja b, turns that amount into a vector by scaling the unit vector in the direction of a: proja b = (a · b)/|a|² a. These projections show how strongly one vector aligns with another and are important in mechanics and electromagnetism.

Vector Projection and the Dot Product

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

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

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...

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 the dot product links vectors, angles, direction cosines, and projections, with clear formulas and key uses in physics and math.

Vector Projection and the Dot Product | 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

Vector Projection and the Dot Product

Description

Transcript